Hey, a fellow Londoner here
Just so you know, city farming doesn't really count so you've got the right idea looking at actual farms(me and my friend volunteered on one and Liverpool told us it didn't count as farm wex).
It comes down to how much travelling you are prepared to do or if you have any relatives to stay with. Sadly, I am not in the second category, as pretty much my entire family lives in London, so I had to travel. My furthest placement was 1 1.5 hours away.
In the end I managed to get dairy, a small holding farm and lambing booked for April as well as volunteering at a RDA. I also got a equine and mix practice vets.
I too have a very nice mum who drove me to my places, but if you don't have that option look for places near train stations.
I found all my placements using google,
Good luck!