"Does the solicitor pay the fee straight away or does it sit in the client account for a few months?"
Yes I keep all of my conveyancing files on the long finger so that I can keep hoards of money in my client account so that I can rake in all that lovely client account interest.
There is just no point in doing that - the only good file is a closed, completed file. And if I did not get the Land Registry funds in advance of closing do you think I would stand any chance of getting them from my clients months afterwards? Particularly if there is an issue to be resolved?
mf