Hi Dawn,
So you're going to "Proms in the Park", and I hope you have a great time.
Hyde park (and the Royal Albert Hall) are outside the congestion charge zone - which only applies on Monday to Friday (working week) - so no undue worries if you want to take your car (other than the cost of parking)
Both Hyde Park (and the Albert Hall) are quite near Paddington and Marylebone main line stations - which serve the West of London.
Ray mentioned some CC sites to the West of London in Buckinghamshire - Seer Green and Chalfont St.Giles - living in Buckinghamshire I know that area pretty well and used to buy eggs from the chicken farm that is now the commercial site in Seer Green (no chickens any more though). These sites have easy access from the M40 and M25. Train from Beaconsfield or Denham will take you into Marylebone - Seer Green train station is a "halt" and only open during the rush hour.
Welwyn Garden and Hertford are to the North of London (and certainly aren't 20 minutes to London) - train will take you into King Cross or Liverpool Street and you'll need a tube journey around central London to the West.
Chigwell is further round to the East.
The two Caravan Club sites in South London are good, inside the M25 and so have access to underground and local transport services, with reasonable access from the M25 - but I'd pick somewhere to the West.
Robert