Incorrect name on register

Over 20 years ago , when we bought the property, clearly identification checking wasn't as thorough as it is now. My husband's official first name Is Edward but he has never been known as that, and has always used his middle name instead. That is the name on the register and now we are selling, the purchasers solicitor has raised the question (quite rightly) How much of a problem is this going to be?

Angela - it's rarely a problem if all parties carry out the required checks and confirmation re legal names and names used.
Clearly he needs to prove that he is who he says he is and that he is the person as named on the register. He can't do that using his passport/birth certificate on their own presumably as he 'used' his middle name when buying the house and the conveyancer allowed that to happen. They would still have verified his identity at that time but as you suggest maybe the checks carried out then were not as rigorous as nowadays
Your own conveyancer can check and confirm that the person named on the register and the person selling is one and the same. The buyer's conveyancer may be willing to accept that but often it's the buyer's lender who is the stumbling point as their process flags such a mismatch and the process halts 
If the conveyancers can't satisfy the ID/mortgage needs and accept that they are one and the same person, confirmation of which is then included with any application to register the purchase, then the only option is for your own application to be submitted to update the register re the name before you exchange/complete
So if it is a problem for the buyer the solution lies in your hands to remove that problem 
If you do have to apply first to update the register then that can be submitted and a request to expedite it then submitted afterwards. That will hasten the processing in light of the linked onward purchase