What happens if one of us applies for divorce outside our country of
birth?
Are You Married?
If one of you applies for a divorce outside the county where you got married, the court in the country where you are applying for the divorce will first ask: "Are you legally married?"
The court will look to the country where you got married (and any other country where you lived after you got married) to determine how to answer this question.
The court could (1) conclude that you are married because it finds that your were legally married in the place where you had your religious marriage for its purposes, or (2) the court could find that when you moved to some other country before moving to the country where you are seeking divorce, that the intermediate country recognized "common law marriages" (in a common law marriage country you are married if while living there you hold yourselves out to the public as married, view yourself as married to each other, and have consummated the marriage or cohabited), and that you became married then, even if your original religious marriage wasn't a legal marriage. Common law marriage is recognized in some U.S. states (including Colorado, where I live) and existed in Scotland prior to the year 2006.
I personally am not able to clearly tell from the facts in the question whether or not you would be considered currently married in Australia, Canada or NZ based on the laws of the country where you had your religious marriage.
If You Are Not Married
If the court determines that you are not married, your property will be divided and custody of children and child support will be determined as it would be between unmarried people in the country where the application for a divorce was made and a divorce will not be granted because the court will find that you weren't married in the first place.
When people are not married, property is usually divided based on formal legal title to property, usually there is no alimony, usually custody is handled in accordance with the best interests of the child, and usually child support is determined based upon local law based upon the economic means of the respective parties and the nature of the custody arrangements.
Some of the important qualifiers to these general rules are that:
(1) some of the jurisdictions in question recognize a concept called "putative marriage" which afford the rights of a married person to someone who isn't legally a spouse but who believed in good faith that they were a spouse that accrued through the time when the person seeking relief discovered that he or she wasn't actually married (often this isn't formally recognized in statutes but is an equitable remedy provided for by case law);
(2) some of the jurisdictions in question give people who have cohabited (particularly if they raised children together while doing so) some statutory legal rights beyond mere strangers vis-a-vis each other that arise from what is sometimes called their "de facto" relationship even though it wasn't a marriage relationship, if court action is taken fairly swiftly following a breakup (e.g., Australia's de facto relationship law is explained here and varies somewhat by province, and Canada has similar laws in most provinces);
(3) some of the jurisdictions in question do not recognize putative spouse rights, or de facto relationship rights, but will honor verbal or written agreements between two people in a domestic situations upon which the person seeking relief relied or for which the person seeking relief provided some benefit to the person from whom relief is sought, under contract law. But, if the sole benefit provided in exchange for the promises was sex (this is called a "meretricious" relationship) then the contract will not be enforced on the grounds that it was an illegal contract (this is usually a case law development interpreting existing contract law, rather than a statutory legal doctrine).
If You Are Married
If the court finds that you are married, then a divorce proceeding will follow according to the same laws that would apply if you had originally been married in the country where you are seeking the divorce (except that if you lived in a country with community property when some property owned by the couple was acquired that community property will be treated as owned 50-50 by the two of you regardless of how that property is titled).
Typically, this ends the marriage and also involves an equitable division of marital property, an alimony award if appropriate for an amount and a duration based upon the circumstances, custody arrangements for the children according to the children's best interests, and child support based upon the respective economic means of the parties and the custody arrangement which is put in place.
Also, Australia and New Zealand and Canada all have a waiting period from either separation, or from filing a divorce, before a divorce can be made final in some circumstances.
Country Of Birth Rules
What happens if one of us applies for divorce in our country of birth?
You have stated that:
our country of birth doesn’t recognize civil divorce if the marriage
wasn’t civil in the first place.
and, that you didn't have a civil marriage. But, you don't identify the country of birth which is important because we need to know what that country's laws are when you have a religious marriage which is legally registered but not a civil marriage. The answer is both non-obvious and is not necessarily the same from country to country.
Some countries would say that you are still married in that situation and deny you a divorce but might possibly allow a judge to decree arrangements for a legal separation, or allow you to seek to have the religious marriage annulled if it was in some way defective.
Other countries might treat the two of you as if you had never been married, in which case the law pertaining to unmarried couples would apply, apparently along the lines that you suggest in your question.
Also, some countries (e.g. India and Israel) have different marriage laws for people who belong to different religions. So, it may be necessary to know your religion as well as the country where you got married to determine what would happen in the country where the two of you married.
But, without more details, this question is impossible for anyone to answer accurately.
Property Division Rules
How property is divided especially pre-marital assets such as
pensions, house brought into marriage, saving accounts, inheritance /
expected inheritance, and investments?
Mutual Agreement
All of the jurisdictions mentioned allow the two of your to come to a mutually agreed separation agreement made with full disclosure of the financial facts so long as it does not cheat the interests of your mutual children, in lieu of having a court decide these issues. A mutually agreed separation agreement does not have to divide property in the same way that a judge would have divided the property according to local law. But, if there is not a mutual agreement a judge will decide how to divided the property of the couple
Judicial Division Of Property In Separate and Community Property Systems
All of the jurisdictions mentioned, if they find that there was a legal marriage, or that putative spouse rights apply and that there was not a mutually agreed separation agreement, would have a a judge make an equitable division of marital property following a hearing at which evidence would be taken.
To the extent that the property of the respective spouses is treated as being part of a "separate property system", that is basically it (although there is some case law guidance about what is fair and what is not). The judge is charged with "doing the right thing" as long as it is fair and equitable.
To the extent that the property of the respective spouses is treated as being part of a "community property system" (e.g. the property was acquired in CA or Spain and the court recognizes the community property system status of property acquired in that time period), property is divided into the separate property of each spouse and marital property. The separate property of each spouse stays the property of that spouse and the marital property is divided between the two of them "equitably", i.e. in a fair manner starting from the initial assumption that 50-50 is fair but subject to arguments that this should be modified for some reason (e.g. to provide "lump sum" alimony). In a community property system, property owned before the marriage or acquired by gift or inheritance is separate property, and property acquired during the marriage or as a gift to the couple or co-mingled by the parties with marital property, is marital property. The treatment of appreciation in separate property is a fine point that varies considerably. Pension rights are usually considered property that can be divided if they are marital property.
None of these jurisdictions would treat an expected inheritance as property to be divided, although an expected inheritance might be considered as one of many factors when making an alimony award.
But, given that there are many different possible jurisdictions involved and that there are already a lot of uncertainties in this question (like what property was acquired in which countries), it doesn't really make sense to get to detailed about what the particular property division rules apply.
UPDATE (March 27, 2019):
[Will] a court of law in our new country take into consideration the
laws of our country of birth when dividing property especially that we
entered this marriage while implicitly agreeing that we have separate
property before and during our marriage[?]
Usually not (in the absence of a marital agreement in writing complying with the new country's laws).
But, some places that have separate property will recognize the community property/separate property status of property acquired in a community property jurisdiction.
In the U.S. the treatment varies from U.S. state to U.S. state and is not uniform nationally.
I'm concerned with assets acquired before marriage, the matrimonial
house which was acquired before marriage, the pensions as one of us
has a bigger pay check,
These details vary considerably even within community property jurisdictions.
For example, in some jurisdictions, appreciation in separate property and income from separate property during the marriage is separate property, and in other jurisdictions, appreciation in separate property and income from separate property during the marriage is community property.
The place where the divorce takes place usually controls, although some jurisdictions look to the location of the property when acquired or when owned.
Almost no Western jurisdiction in the 21st century cares which spouse had a bigger paycheck during the marriage. The "partnership theory of marriage" is widely practices and assumes that both spouses contributed equally to income from work and income from community property during the marriage.
In our country of birth, property is what you have your name on it
which requires no domestic contract to be signed before marriage.
This is true in most community property jurisdictions including most with "civil law" systems based on Continental Europe's legal codes.
Also, bringing in a domestic contract now might breakdown our
relationship because the other side may see it a sign of lack of
trust.
True.
I personally don't find it fair to divide an account of a million
dollars with someone who didn't work hard for it and/or inherited it.
In the 21st century, almost every Western country in the world (and many non-Western countries) disagree with you, although many divorcing husbands and ex-husbands do agree with you.
