Yeah, I was going to say that you need to be more specific given different parts of the UK are pretty different and transport varies from reasonable to awful depending on where you are :)
I don't know Essex at all but I do know London and the edges of Surrey from living there while studying.
It depends on when you go but some bits of advice regarding this area:
Central London is basically useless for second hand ponies. It's not great for not second hand ponies either, honestly. You need to be in zone 3 or outwards (that will make sense if you look at a tube map).
There are some decent charity shops in the outer zones, but most areas of the UK will have those and whether they have ponies when you visit them or not is a bit hit and miss. I picked up quite a few pony items in Kingston and Tolworth (both zone 6 I think) but as I said, it's not a guarantee. Just that there are more in the outer zones.
Camden used to have a shop with old ponies (mostly G1) in the market but I think it's gone (at least my sister says it has).
A better bet for you - if you are going around the late spring/summer - is the carboot sale they have in Ewell every sunday morning (when the weather is ok) from around April till October. It's in a big open event field. Putting the details in a spoiler tag so it doesn't spam up the post :D
Spoiler
I don't know if you plan on having a car while you're there and I can't give meaningful road directions anyway, but if you google the Hook Arena carboot sale it should come up. If using transport, you can get to it from central london by overground train (London Waterloo to Ewell West, it's about 30 mins) then walk from the station (it's a pretty straight line from there). I used to get the bus (406) from Kingston but you'd need to get to Kingston first so I think it makes most sense to just get the train from Central London if you don't have a car.
In anycase, this is a very busy carboot sale in the summer. It also runs on bank holiday Mondays (of which there are 2 in May in the UK). You need to get there pretty early (no later than 8 or 8.30) since there are a lot of people looking for deals, but it's probably the best place in that area I can think of for ponies.