In the south east it will cost more- but Kent will probably be a lot cheaper than London. It's obviously got a lot to do with how much land is worth, and the costs the yard owner has to pass on. Say 100-150 a month for DIY livery outside of London.
Then on top of that, feed, hay and bedding- again in the south east, the hay and bedding are likely to cost more as the land can be used in more profitable ways, and you're probably not going to be able to have the horse/pony out 24/7- so say £30 for this as a rough estimate (some horses obviously need a lot more feed than others).
Say £60 every six weeks for the farrier- again a working average, some people will pay less, some people will pay more.
Then maybe £25 a month for insurance (obviously not everyone takes this out, but otherwise vet bill can run into £1000s). This can be a lot more depending on the history and value of the horse and exactly what it's insured for. Then maybe add another £100-150 on for vaccinations and dental treatments (this can be less or more depending on the vet and call out fees etc, and not everyone will bother with flu vacs/dental).
Then you have the one off/occasional costs of buying tack/rugs/other equipment.
And then lessons/competing/transport will be an additional cost.
So say £250 a month, but this is a massive average and some people will manage with a lot less and some people will spend a lot more- but that is doing all the work yourself and not really paying for any extras/non-essentials.
It's also worth bearing in mind that knowledgeable people with access to their own land, or maybe just renting a field can keep that cost down really low, whereas newer horse owners who maybe want a bit of support from external people and maybe need to call the vet more and want an instructor around every so often and maybe feel the need to buy more stuff (and have less stuff just lying around that they can use) will end up spending more.
And as others have said, it's very easy to end up spending every spare penny on a horse!