This is a how long is a piece of string question because assisted livery can mean different things at different yards. You need to have in your mind what help you need and then ask various yards what price they would charge to cover your requirements.
In this area expect to pay anywhere between £250 and £450 a month depending on what assistance is included and whether bedding/forage is included. That's based on looking for assisted DIY in the Surrey area just before Christmas.
Mine is about 400 I think? That’s including a large stable, feed, hay and bedding included, a few acres of turnout with the other liveries and access to a big sand school. What’s not included is the mucking out, poo picking and evening bringing in (tho they rug change/turn out in the morning). I think that’s on the cheaper side for what I get. However I would think it varies massively depending on how close to London you would want to be - don’t know about part livery but around Kingston and Wimbledon I know full livery can easily be a thousand per month 😮. The cheapest places are generally the furthest out into the countryside, so it just depends on where you are really.
It really depends on what assistance you want. Some yards encourage buddying up so they don't offer assistance, but it is there in another form. I looked in December too and found between £150-£300/month basic DIY then with assistance charges, bedding, feed and hay on top. Further out isn't always cheaper I found, but horse numbers per acre is often better further out.