Oyster3 Stroller

I had the same criteria as you and I got an Oyster 3. It completely fit the brief and I loved it. I got it for my first "baby" who is now 5 and a half, then used it with my next one too, and then gave it to a friend to use when she had a baby, and now I'm expecting my third in November and so she's giving it back to me. It's still going strong! One other thing I really appreciated about it was it has a really good sized basket to stick all my shopping/rucksack with baby stuff/etc in. One feature that I literally never used was attaching the car seat to the buggy. The car seat stayed in the car on the Isofix base, and babies aren't supposed to sleep in car seats for too long anyway, so if my babies fell asleep in the car I would always just transfer them to the carrycot if we needed to get out. It looks as though BabyStyle have a new model out now called Oyster Gravity+ 4, which I can't vouch for except on the basis that the Oyster 3 was so great. Might mean you can get a good deal on the 3 somewhere?

Babies get a lot of early social development/emotional regulation by being parent facing from birth. I know I smiled and talked to my babies a lot more because they were looking at me. We've used an Oyster 3 (secondhand off Facebook marketplace, and got replacement parts when they broke from ebay) and it's not compact but the seat can lay flat and change to parent/world facing. It fits flat if I take one wheel off in my Peugeot 208. It looks like a Yaris is a similar size boot.

we also love our oyster 3, and planning to use it again for baby 2 due later in the year!

FTM and absolutely adore my oyster 3! Super easy to use, folds quite small and feels more sturdy than some of the really popular brands.

We also have and love the Oyster 3! Got it for our first who is now 6, will be using it again for our second baby due this summer.

We used the Oyster 3 and used to carry baby upstairs in the carrycot often. It is a sleep safe carrycot (watch out for that; some are not necessarily intended as a proper sleep space though they're fine for little supervised naps, so think about whether you might want to use the carrycot as an actual second sleep space - we did when travelling). I don't live in a flat but our house has the living room and kitchen upstairs so we'd usually want to take the baby upstairs when we get in if still asleep. It can also be rocked gently while on the floor which can help soothe baby back to sleep if they start rousing with the lack of pram movement. However, I'm not too sure that I'd go for this one if i were you living in London, as it's not the most compact travel system. You could see if you could find it on display somewhere (or I think it might be oyster 4 now) to see what you think. Alternatively some prams have "cocoon" attachments, only for little newborns but they are great for carrying sleeping babies. I now use a Mountain buggy duet (because I have a baby and a 2 y/o) and it can be used with either carrycot or cocoon, both of which are suitable but the cocoon is probably easier for your purposes. You obviously don't need a duet but there are single prams with this option too like the mountain buggy nano.

I have the Oyster 3 and have loved it - I'm still using it for my second baby now, so it's withstood heavy use for 4+ years. It folds fairly flat and is lightweight to push around - other people have also commented on the latter, compared to their own prams. I got the bassinet as well, so use that as well as adaptors for a Joie car seat. I don't know anything about the Oyster 4 tbh, but if it's like the 3 I'd say go for it!