Title registration

We are purchasing a property with a strip of land next to it that looks as though it is within the boundaries of our home. (Closed off with the rest of our garden) On advertising they said it was a possessory title over this bit of land as the owner has maintained the land for over 50 years whilst owning the home. 
Our conveyancer has requested the vendor register it as an absolute title in there name before completion but they are refusing saying it will take too long and for it to be a possessory in our name and they’ll provide a statement of truth so we can do this afterwards. 
We have since found out it is in-fact unregistered land. I was wondering how long it is likely to take if expedited to get this land registered. I was under the impression it may take some time and would have been quicker if It was a possessory title. 

Aimee - two things to comment on although you should rely on your conveyancer to advise/assist you as to the options and 'best' way to approach this.
The vendors are only likely to get absolute title if they make an application to include the land claiming it was an error on registration and/or providing deeds to confirm that error for example.
If they couldn't do that then their application to register would be the same as your own and based on the facts as claimed and as stated in your post. Invariably only possessory title can be awarded in such cases.
The quickest way is if there's been an error but it doesn't read as if that's the case.
If the vendor's apply then they can request expedition to keep the wait time short but it will still most likely take 2-3 months to process providing everything is in order and falls into place. The application will need to be in order; a site visit/survey requested by us; and wider checks with adjoining landowners carried out in most cases - that all takes time
But the most important thing to stress here is that if you are buying with a mortgage then the choice may not be all yours re how to proceed.
Your conveyancer can confirm/clarify as appropriate so please do rely on them for the best way forward here