Sorry you're going through this. It sounds like it's been a shock to you as well, which doesn't help.
Firstly, it would be sensible to set out some ground rules for this initial time period. What you are going to do about telling the kids, how you will continue to coexist and how you will behave towards each other. If you don't work then you should also agree what you will do about your finances during this time and how long you both expect to maintain some sort of status quo.
If you're both sensible enough about it and you can do so calmly then you could start to open up on how you see co-parenting working in the future and how you would like to deal with custody.
You need to start doing some research, get on Moneysavingexpert and work out what you're entitled to, both in terms of any benefits and also child maintenance. Then you'll need to consider your options in light of all of the above.
Have you paid any money into the house (notwithstanding that it's in their name)?
There will be a lot to think about and at this stage try as hard as you can to stay civil and cooperative. Assuming that by 'partner' you mean unmarried partner then you don't have a lot of rights and so it will be helpful.