Forget about the car and flat. You can cope in a small car, you'll have to leave the car seat in situ and just put the baby in and out. Take the largest room in your home and turn it into a bedsit for yourself and partner. Baby gets the smaller room for their bedroom/toys.
One or both of you will need a new job. A great many people, usually women, lose their careers due to having a baby for the simple reason they can't afford to return to work until child is in nursery/school, then are limited in the hours they can do. This is often but not always due to men refusing to compromise their career progression at all so their partner can keep her career to some extent. The other factor is women wanting a few years as a SAHM over and above all else in her life. And lack of informal childcare or ability to pay for formal childcare, children have school holidays or teacher training days and they get sick, employees only have a certain amount of annual leave and often it doesn't stretch far enough, so one person ends up part time or not working.
Can you live solely on partner's salary, without him being financially controlling and considering it "his money"? If so, returning to work and spending all your wages on childcare means you earn zero, for now, but also means you keep your career and at some point in future you will stop paying it all out in childcare. It means if you split up you have your career still, this is weighed against losing those first few years with your baby as a SAHM and still effectively earning nothing. It's personal choice. Either way, money needs to go into a joint account (both sets of wages) and all bills paid from there, including childcare and some into savings for emergencies and pension contributions for you both, all big purchases talked through first by both of you, all money is family money from now on. If there's any left over you can divide it equally (either drawn out in cash or moved into personal bank accounts) and that's your personal spends for the month. If he won't do this, consider becoming a single parent (ie you live separately) because you may be better off (and claim child support from him).
Go on Entitled To website and enter your hypothetical details as they would be a) if you stay as you are once baby born, b) if you become SAHM once baby born and c) if you're a single parent. This will give you some idea of benefits you could claim in each of these situations and can help you decide what to do next.
Sit down with partner, draw up a budget and don't miss anything out. If you're left with a minus figure at the end of the month, you'll have to get really ruthless about what is truly an essential and what is not.