Oyster3 Stroller

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 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!

I got an Oyster 3 secondhand which has been great - the seat lies flat so can in theory be used from birth, though I'm not sure the straps really go tight enough for a proper newborn if you don't want to use a carrycot at first.

Although it will cost more than a single package, you might consider getting the Joolz Aer+/Aer² for your main pushchair and then buying a second hand Out n About Nipper Sport v5 for the countryside. The sport version is cheaper than the normal version as it has a fixed front wheel. Both lie flat so can be used from birth, although you shouldn't run with the nipper until your baby is 6 months old. The wheels come off the nipper for putting it in the car (though if you've got them muddy then you might want to leave an old towel in the boot for a quick wipe down). A true do it all solution doesn't exist which is why many people eventually get a compact travel pram in addition to their main day to day travel system. A bassinet/carrycot attachment can also be absolutely worth it, even if your baby grows out of it in 4 months. Many are rated safe for overnight sleep so you can use them in the day time for naps in the living room, or just as a safe place to put baby while you do something, without having to take them to or move the cot in the bedroom. Some babies like carriers and some don't. Same for pram bassinets/carrycots. You won't know until your baby arrives unfortunately. I personally bought a Babystyle Oyster3 (with carrycot) and love it. It's a good all-rounder but probably bigger and heavier than you want for city living. The Oyster4 has now been released and has a much better folding system. I think it's mid range on price but does everything you need it to so I'd recommend it over more expensive options. Maybe it's worth checking out if you only want to buy one pram.