There is a possibility that you would be able to register the prescriptive right, however, I would suggest that you have the seller do this as they would be the ones with the required evidence to make the necessary application to HM Land Registry.
The basic principal of prescription is that, if a land owner has exercised a right over his neighbours land, nec vi, nec clam, nec precario “not by force, nor stealth, nor licence” and for a long period of time, the neighbour has effectively lost their right to object to that right and has acquiesced to that right being made lawful. A claim to register a right of way is often made on the basis that a right of way has been continually used for a period of time in excess of 20 years or on the basis (or sometimes fallacy) that the right is the result of a historic grant.
Therefore, the existing seller would be in the best position to register the easement.
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If you have any further questions, please do not hesitate to ask.