Here is a shortlist of sites that had availability next weekend on Wednesday - none of these was exactly what we were looking for, but may suit you? Although I think they are all probably further than you would like. Prices are for 2 adults and 2 kids - comments were aimed at my friend based on what we were looking for.
March, near Cambridge - Fenland Camping -
http://www.ukcampsite.co.uk/sites/details.asp?revid=4449 - £11/night inc
EHU, 2 nights (I think) only downside is no football allowed but is very
cheap as I think they will allow a 2-night stop and looks pleasant enough
Uffington, Oxfordshire (where the white horse on the hill is) - Britchcombe
Farm - http://www.ukcampsite.co.uk/sites/details.asp?revid=4476 - £12 inc
EHU, 50p showers, 2 nights - they commented no campfire as pitches are near to the buildings (on their other pitches they provide firewood for campfires)
Poole - South Lychett Manor -
http://www.ukcampsite.co.uk/sites/details.asp?revid=1233 - £25/night with
EHU 3 night min - all new facilities but well pricey!
National Forest, Derby - Beehive Farm Camping -
http://www.beehivefarm-woodlandlakes.co.uk/caravan_and_camping_site.html -
£15 no EHU, 3 nights min, few toilets according to reviews but might be
improved
Henley - Swiss Farm International -
http://www.ukcampsite.co.uk/sites/details.asp?revid=1366 - £17 no EHU, 3
nights
Hope this helps
Christine