The Land Registry has a mechanism where a right of way can be registered directly with them if all parties agree to the right of way's existence. It's called a Form 68. That might work out faster than getting it confirmed in the District Court but I'd recommend ringing the Land Registry to see if there's a backlog on them at present.
Arguably, the seller's solicitor is being over cautious. If their client accepts the ROW exists then things can continue as they currently are after the sale. If they would prefer the ROW wasn't used (many people's preference) they have a better chance of that happening if it isn't registered on the deeds and when the elderly neighbour passes, the new property owner may not be interested in using that backyard access.