In england-and-wales the financial position between the partners could end up being resolved in family court, rather than as part of a civil claim. Bigamy can be taken into account in the financial settlement during nullity proceedings (similar to divorce) and the bigamous partner may end up with less than they'd normally expect from the breakup of the relationship.
Suppose A marries B and later "marries" C while still married to B. The A-C marriage is void. A court, probably on application by C, can formally declare it as such, and deal with the division of marital assets in the same way as in a divorce. (This was a 1970s reform intended to give fair treatment to people going through an annulment, potentially after a long relationship where they assumed that they were properly married.) There is a doctrine that the court would not assist A in gaining a financial advantage from his unlawful behavior, which applies in this scenario to the extent that A is culpable for the bigamy and C is not. That's not to say that A will get nothing out of the split: the starting position is still a fair division. But it can block A from claiming support out of the marital assets to which he might otherwise be entitled.
This issue arose in Whiston v Whiston [1995] Fam 198 in the Court of Appeal: Mrs Whiston was the knowing bigamist and she was denied the £25,000 she wanted in ancillary relief. Since then, more recent authority has clarified that the doctrine is not automatic, but depends on the context of what A did (Rampal v Rampal [2002] Fam 85). The crime of bigamy requires a certain level of intention (the "mens rea") which is not the case for all people: sometimes, partners are genuinely ignorant of the state of a preceding marriage. Further, if C knew about the bigamy all along (as in Rampal) then she can't take advantage of the situation years later. That has been a general theme in some of the recent cases, often involving an A-B marriage contracted in another country under unclear circumstances, swiftly abandoned but not formally dissolved, and then an A-C marriage in England which lasts for many years until C decides to weaponise the situation.
A very recent case PF v QF [2024] EWFC 10 casts some doubt on whether the remedy in Whiston is still available, given other developments in family law, but for now it still seems to be in play. PF is another example of where the doctrine was invoked but ultimately not agreed to (97-98):
I do know that both parties worked during the marriage, and cared for their children, and overall there appears to have been what is these days a fairly standard pooling of assets and resources, and a sharing of responsibilities. It is difficult in those circumstances to characterise the applicant’s claim as an attempt by a bigamist to enrich herself in a way that would not have been possible in the absence of a fraudulent or criminal act. [...] For those reasons, this claim is not, in my view, an abuse of process. On the contrary, there are clear public policy arguments in favour of recognising the partnership and ensuring a fair division of its assets.
In different circumstances, where there was an attempt by a bigamist to enrich himself in a way dependent on his fraudulent or criminal acts, the outcome could be otherwise. A further note is that this line of cases all have to do with when A's claim can be struck out (effectively that it's denied automatically), but the bigamy issue could also arise at a later step, where the judge can adjust the settlement in the light of all the circumstances including A's conduct. See the Matrimonial Causes Act 1973, s.25(1) and (2)(g).
In scotland there is much more limited scope for bringing A's conduct into account during the financial part of the process. This is generally limited to instances where A had somehow dealt unfairly with the assets in question. It's possible that a similar idea could be argued as in England, in the right circumstances, but so far it's untested.
The marriage statutes on this topic in northern-ireland are very close to the English models, and so if this issue arose there, it's plausible that the local judges might follow English precedent. But again, since these are smaller jurisdictions, there's less definitive case-law involved.
