Unless you're very strong you won't want to be carrying your baby in a car seat up & down to the flat, they're very heavy!
Everyone's different, but if I was having a baby now & living in a flat, I'd get...
A travel cot (one that has a high position (then medium & low) for the lounge and some form of side-by-side for the bedroom
(You should have the baby sleeping, even in the day, in the room you're in).
Then later on a decent size cot bed. No need to get it before you have the baby though!
The travel cot will come in handy for (U.K. holidays/visits to friends & family, obviously) but also when they're crawling to put them in while you wee/shower/answer the door/whatever)
Bath depends on your back!! Whether or not you have a bath in the flat etc. I've had a variety from the bucket shape to the full bath shape (went into duty as lots of things afterwards!! Laundry basket/toy storage, outdoor play) for a while I used it on a table when my back was bad.
Changing table... I'm short, so the ones that go across the cot so you're at the feet don't work for me. I preferred having one across a chest of drawers! (But also just an ordinary squishy one in the lounge for on the floor) [these days my back is wincing at the thought!!] but given it's a flat, you might be fine just using the one in the bedroom!
Definitely a carrier like the one in the photo, or a sling. Some people really do well and can carry older children in a sling (but not me).
Baby bag - one you like, because you'll be carting it around for a long time. One that DH is happy to carry so you can palm it off on him!! Depends if you travel light (one spare nappy) or like me, go prepared for all eventualities!! Doesn't have to one sold as a nappy bag, but generally they're good because they're lined and the straps are long enough to fit over the handle bar of the pram etc and are spacious with pickets & internal bottle holders etc. not essential, but generally handy!
High chair, either antilop IKEA -dead cheap & easy to clean, or a trip trap more expensive, but can carry on being used right through to teens fir an extra chair at the table, at a desk or for instrument practice. (And you can get small baby attachments)
Pram - this does not have to cost as much as a car!! Be warned though, they take up FAR more boot space than you think they will, so check before you buy!!
Car seat. Buy new. If I was buying one now I'd look at the birth to 4 (or older) rear facing.
Small things:
Muslins
Vests (look up how to remove when covered in poo!)
Baby towels (obviously not essential, but very cute and easily identifiable so don't get used by accident (repeatedly).
Linen for whatever you decide for sleeping
Nice things for you!! Good quality 'travel' cup.
Thermometer
Don't be overwhelmed, just enjoy it!!
Xx