For example, say I were to have a contract with 3 others. And all the requirements being met, we would have a joint tenancy in a property. If we (landlord included) were to amend a new party to the tenancy agreement, would this negate the unity of time requirement for a joint tenancy, and would we therefore have a tenancy in common?
You are confusing two different concepts.
Despite their names "joint tenancy" and "tenants in common" are ways of owning property: not leasing it. Your landlords will be one or the other (or "sole tenant" if there is only one of them), you and your housemates are simply "tenants"
What I think you are driving at is the concept of "joint and several liability" where each tenant is responsible to the landlord for the acts and omissions of all the others, including the omission of not paying the rent. That is the implicit condition, however, the parties are free no negotiate whatever they like, that is, it is a creature of the particular contract(s) between the tenants and the landlord.