Simon - what they have advised re the land swap is what's legally required to transfer ownership.
What's required by a solicitor or perhaps your mortgage lender is more about 'risk' assessment than the legalities as how and when you exchange/complete is a conveyancing/lending decision so not a 'legal' issue.
So if they need a 'clear title' before they will complete then the application(s) to register the swap need to be submitted. They then have to
request expedition to keep the wait time as short as possible. Everything then hinges on the applications being in order of course.
There's no government UKLR and that relates to a third party service provider. If the land/property is in England or Wales then you would be dealing with us at HM Land Registry
And there's no 'All' on the forum as whilst others can comment they very rarely do as we use it very much for HMLR to answer enquiries only