Hi Ishan - that's interesting advice from your solicitor and whilst wait times can be lengthy the commentary/advice given is not the type I have come across previously.
As you already appreciate we have an expedite service that hastens applications where there's an urgency involved such as a linked/dependent transaction.
Request an expedite - GOV.UK
A remortgage would be such a transaction
Fixed year mortgage deals are the norm for many but there is zero reason in my experience or from a registration perspective to be concerned re the wait time referred to. And certainly no point in requesting expedition now when there is no urgency as the request would be rejected.
Yes, speak to your broker/lender as appropriate but don't shy away from a 2 year fixed deal if it's the right one for you both. And just make a note on your calendar to check to see if the registration is complete 6 months before the end of the deal and/or when you are ready to start the remortgage process. Once you have a new mortgage offer, and IF the registration has not been completed, you can then request expedition as appropriate