Can just tell you what I have The Loola Up. I love it, I think it's fab, it doesn't seem to get recommended much, possibly because the Loola is supposed to be a bit rubbish, but the Loola Up seems to have solved all the problems with the Loola. You could also try looking at the Streety or other bebeconfort pushchairs.
- light enough to get over a steep hill
I live on a steep hill and it's not impossible. I think it is slightly heavier than some other buggies but the wheels turn very easily. I find it much easier to push this buggy up the hill than walk up carrying something of medium weight
- not too big for a bus (I don't drive)
It fits easily on every bus I've tried - I don't drive either. But if your local buses require you to fold it first, it might not be a good one as it's quite tricky to fold/unfold one-handed holding a baby (though if you were using with a carrycot, then it wouldn't be a problem as you could put the baby down, and it is very quick to fold with 2 hands)
- reasonable size shopping basket that can be accessed easily
Reasonable size, yes, also can hang a changing bag on the handle unless the handle is in its highest position. There are carrier bag hangers on the handle too. It is easy to access in forward-facing mode, on rear-facing mode you do have to walk around to behind the seat or reach around the side.
- choice of rear and forward facing
Yes, also has 4 recline positions both ways, you can recline/sit it up with the child in it, it saves the position when folded, and you can switch from rear-to-forward-facing really quickly and easily, almost one-handed!
- not tyres you have to pump up
Tyres are smallish and not air tyres.
- not too big when folded down (other kids' stuff in car boot too, needs to be stored indoors at night)
It folds umbrella-style, also you can remove the seat (one handed) when folded or before folding if you can't fit it in somewhere.
- adjustable handle height as I'm tall and get back pain
Handle is adjustable to 3 positions (you'd have to try this out I think in a shop)
- not one you have to buy with a car seat
It doesn't come with a carseat. You can buy a compatible one, but the pushchair seat part lies 100% flat for a newborn and goes rearfacing by itself, so no need to use it with a carseat. If the car seat you have is maxi-cosi you can buy an adapter to make that compatible too.
You can buy a compatible carrycot, I didn't bother because the seat unit lies flat but the reviews I've seen said the carrycot was lovely and looked very comfy.
Price was around £220 when we bought it which included the chassis + seat unit, (including reversible seat cover) hood and raincover. I also liked the fact it has 2 modes of suspension, one smooth for newborns and one for more easily getting up kerbs etc. You can set the wheels to swivel or go straight. And I love the fact the seat cover is reversible, it feels like I have a new buggy every 6 months - the summer side is wipe-clean as well and breathable whereas the winter side is more snuggly. Also as you have older children - you can push it one-handed as it has a bar style handle rather than 2 separate ones.