Though I live in the U.S., I am a member of the RHS. They run articles regularly in their members' magazine about RHS display gardens, some of which are potagers. Try an internet search on RHS display gardens, Potagers in London, etc.

if in London, go to Kew gardens http://www.kew.org/
not what you are after but great anyway, also look out for Garden safaris, where local people open up their gardens for charity, they will have 20+ home gardens to nose round, mainly June/July.
http://www.redcross.org.uk/opengardens_searchresults.asp?id=54243&txtPageNo=1&txtEventLocation=EC2Y9AL

http://www.ngs.org.uk/gardens/local-to-you/south-east/visit-ngs-open-gardens-in-kent.aspx

http://www.kentgardenstrust.org.uk/visit.html

http://www.touruk.co.uk/gardens/gardens_in_kent.htm

http://www.nationaltrust.org.uk/main/w-vh/w-visits/w-beta-map.htm?quicksearch

Most of the above links will include places with veg, but they don't advertise the fact, as they tend to draw the visitors with flowers. try e-mailing, I know Goodnestone park gardens has veg, as I only lived 1/2 mile away and used to cycle there and have a pint at the Fitzwilliam Arms Pub! my the prices have gone up in 5 years! on their map it doesn't even indicate the veg, but it is next to the house, by the greenhouses! and Sissinghurst gardens have just opened a veg garden.
village fates, you will get to see local produce by home gardeners and they will be around to talk to, as they will be keeping a check on their produce before judging! those British Murder Mystries have their heart in village fates!
Hope this helps and enjoy your visit

Can you guess I am from Kent? :) the furthest points of Kent from London is about 70-75 miles, and the trains are not bad, bus services again pretty good, but most of these places are easier to get to by car.

Here is a link that might be useful: RHS

Thanks so much for responding.I had pretty much given up after a couple of weeks and stopped checking - but glad I did today!
I sure will go to Sissinghurst - it is already on our list of places to visit.
I'll take your tip to call first. Goodnestone Gardens for example, has a map of the garden on it's website and even that does not show the kitchen garden. The web site just says that the walled gardens (once kitchen gardens) have now been converted to flower gardens?
We are going to Stourhead too.
Anything in the Lake District/Scotland that's a 'cant miss'?