I'm not a solicitor myself but I've been involved with enough property transactions over the years to know that the conveyancing process involves quite a number of detailed steps and an awful lot of potential pitfalls. People sometimes think that if there's no mortgage involved that somehow the conveyacing process is merely a matter of signing a bit of paper somewhere. It's not.
The amount of time you'd need to spend learning how to complete each step correctly would be enormous and you still wouldn't have the benefit of an experienced person to make sure you're doing it right. So while it's probably legal that you can do it all yourself, I wouldn't recommend it. You can research the internet and find out how to perform a hip replacement yourself if you want to, but I would recommend that either.