Ironically, the chances of actually having domestic conveyancing done by an actual solicitor is pretty small - it's mostly done by paralegals (domestic conveyancing is very much at the low end of the legal market - with corporate at the high end).

Edit: Not to say that domestic conveyancing can't be complex but the fees are so low for the actual conveyancing part (rather than the sales part) that you are unlikely to get a very comprehensive job done. I've heard utter horror stories of what can happen with conveyancing when it goes wrong....