Baby car seats these days are a bit of a minefield especially the Iso fix bases, they accommodate car seats as the child grows. You will probably be better off having one of these that will fit the baby seat, you can transfer easy from car to car. It will need to match the parents one for easy use.
They are a bit expensive, but you will be using it for a long time. Have a look at John Lewis for info.
But I can recommend the buggies that fold like a back pack. They are smaller and quite robust.
When our GD was born 6 yrs ago we bought a stick buggy to use, but now these newer "back pack" type buggies are great value and so easy to use and so compact.
Both our DD and DS have them for their younger ones. My DS wife found one on ebay for a silly price, have a look there.
Enjoy your time with the little one.