He owns the property outright via family money
"Family money" as in HIS family bought the house and gifted it to him or he purchased it with an inheritance? In which case neither of you has personally contributed to the purchase of the house as there was never a mortgage.
Or by 'family money' do you mean your combined household income was used to qualify for a mortgage but your name wasn't put on the paperwork and deeds? I don't even know how that could happen.
If the first, then you have never contributed to the 'ownership' of house itself, only the normal running costs and upkeep. As has he. The fact that you did more childcare, etc etc means nothing. That was a decision you made. You could have demanded 50/50 for household responsibilities, but you didn't.
If the second, and your income was included in getting the mortgage, then you should have insisted your name go on all the papers and deeds. If you didn't, that's on you, too.
The above is on the 'current' house. So no, I don't think you deserve an ownership interest in that house.
As far as going forward and the 'new' house, you will be living in the house and you should pay for your share of the monthly household running costs. But if there is going to be a mortgage on the new house, no I don't think you should pay towards that since you will derive no benefit from that house if it should be sold. If he wants you to pay a certain amount that he will then use towards HIS mortgage, then I'd insist on some percentage of ownership. It shouldn't be 50/50 since he'll presumably have the money from the 'old' house to put towards the 'new' house, but based on what your actual contribution would amount to.
If you will not be on the 'new' house papers and you want to stop paying for repairs, home insurance, or improvements that only benefit HIS home ownership, I'd say that's fair since you don't stand to benefit from repairs or improvements when/if the house is sold, either. And perhaps its time to insist on a fairer split of household duties to allow you to earn more so you can start saving for your own place or at least have a 'safety net' fund in case your relationship goes tits up.
Of course, if you tell him you aren't going to pay towards the mortgage, he could decide to start charging you rent.