A bit late to this thread but let me try and help.
Reception - Located on Flood Street - Closest Tube station is Sloane Square (Circle/District Lines). It's a 10 min walk (by yourself) or about 12-15 min (with child). There are buses that go on Kings Road but I'd recommend walking through the inner roads. Getting to and from Flood Street by car/bus is not ideal due to traffic on Kings Road.
Y1-Y4 - Cadogan Gardens - This is a short walk (<5 mins) from Sloane Square (Circle/District Line) so is very convenient on the Tube.
Y5-Y8 - Hans Place - Closest Tube is probably Knightsbridge (Piccadilly) but walkable from Sloane Square as well (9 mins maybe).
Children come from all over Central London but I think that anything on the Circle/District line is fine. Victoria/Westminster is a popular location.
These is also a school bus service and it's worth checking with them for morning pick-ups. Not sure if this is still up to date but the pick-up spots were as follows -
West Run
07.20 Bravington Road, Maida Vale, W9 3AL
07.25 Lanark Road W9 1QR
07.35 Queensway, W2 5HL
07.50 Strathearn Place W2 2NG
07.52 Albion Street, W2 2AS
08.00 Park Lane at bus stop near the junction with North Row
SW 1 Run
07.05 Outside Temperance Pub, 90 Fulham High St SW6 3LF
07.13 Harbord Street, SW6 6PL
07.15 Fulham Palace road, SW6 6HY
07.32 Jersey Court, Dairy Close SW6 4HB
07.36 Moore Park Rd SW6 jn Cedarne Rd
07:40 Moore Park Road, SW6 2JB (between Fulham Road and Kings Road)
07.45 Chelsea Harbour, at roundabout on junction of Townmead Rd and Harbour Ave on the NW side heading back towards Kings Rd
SW 2 Run
07.25 Outside Warren House, Beckford Close, Warrwick Rd W14 8TT
07.32 Post Box next to
near 117-133 Oakwood Court, W14 8LA
07.40 Campden Hill Square W8 7JY
07.45 Walnut Court, St Mary's Gate, Kensington, London W8 5UB
07.55 Queen's Gate Terrace SW7 5RR
Don't know whether they still do evening drop-offs as there wasn't as much demand previously.
Hope this helps.