www.addisonlee.co.uk
Book with them, booking a Standard Car and ticking the box saying you have a baby. You then automatically get a VW Sharan people carrier with a fold-down baby seat. These things carry a lot of stuff ... we got 4 adults, 1 x 2 year-old and luggage into one at Easter.
99.99% reliable, they are the biggest car service in London by a factor of about 10 and use only brand new vehicles.
Cost from Heathrow will be about £70 (going back is cheaper as you are not paying the car parking fee).