But if the OP is being subsidised by him (which most women on mat leave are)
No she's not. She's providing childcare while the father is free to carry on earning without that responsibility. They are both equally dependent on the other to facilitate each other's roles.
If the OP and her DP get married, the house could easily be considered a pre-marital asset in any future divorce because it was bought and paid for before the onset of the marriage. There is no hard-and-fast law about this AFAIK, but there is certainly lots of precedents.
So at this stage, it is entirely the OP's choice whether she wishes to protect the likelihood of that asset being kept a pre-marital asset. To do that, she needs to keep it separate to the joint assets, which would include not using any income from it in ways that could be considered to affect the marriage. If she ploughs income from it into the marriage, including to the DP's other child, she is including it in the marriage and giving her DP a potential share of it should they divorce, or 100% of it should she die.
Now if both DC were the OP's and her DP's that would be all well and good - why would you want to deprive your OH and your children of your material wealth in the event of your death? But if you have blended families or children that aren't the product of your marriage/relationship, it is different. The OP could end up with a situation where she dies, her then DH inherits, he remarries to a woman with DC of her own, dies, leaves house to new wife, who then leaves it all to her own DC without either biological child of the DH getting anything, let alone the OP's own child.
Things are different when there are blended families. Our legal system just hasn't caught up with it yet.
OP I think the biggest thing you need to concentrate on right now though is your PND. The way you feel is not necessarily how you'll feel when you're well, and at this stage you can't know how much of what you feel is due to PND and how much is a genuine issue in your relationships. Please be gentle on yourself right now. As long as you are making an effort to be nice and kind to your DSD when you see him, don't beat yourself up about it right now; just concentrate on getting better. Things will all become much clearer then.