Ages of the children?
A40 is notoriously slow many days around Whitney.
A40 has roadworks near Oxford.
What size car will you get that will hide all your luggage and still have room for all 5 of you?
What will you do with the luggage while you are visiting Harry, Hermione, and Ron?
It is M25, M23 to reach Gatwick from Watford. Expect delays on the M25 at least at from the M40 to the M4 and Heathrow, and then near the A3 and the M3, then at junction 9 for Leatherhead and possibly in the run-up to the junction for the M23 and the Brighton Road. - - that is if there are no accidents. Don't offend the gantry cameras if the variable speed limits are in operation on the M25.
Expect slow trucks and cars on the steep hill on the M40 approaching Stokenchurch.
For the train option:
Train from Cheltenham to Birmingham New Street Station by Cross Country Trains is one stop. Change to Virgin Trains West Coast at New Street and you will stop at Birmingham International, Coventry and Watford Junction. From New Street Virgin run three trains an hour to London Euston, one each stopping at Rugby, Milton Keynes Central and Watford Junction, so you need to be sure that your train stops at Watford. Then just out of the front of the station, cross the bus station to stop number 4, and the Knight Bus will whisk you down the road to the Tour.
To Gatwick, back to the station, then train to London where you have a choice. Either walk a few minutes or take a bus one stop to St Pancras International Station and Thameslink train direct to Gatwick. Or Victoria Line tube to Victoria station, upstairs and into Victoria Station, and then either Gatwick Express or Southern train direct to Gatwick (nonstop on Gatwick Express or one or two stops on Southern).
As you see there are significant difficulties with both trains and cars connecting these three well separated places.
