There are two ways of dealing with this that I know of and I think you may be confused between them.
If you are claiming a right to the unregistered land itself, the process is by 'claiming adverse possession'. The period of time is a minimum of 12 years and the period of time is additive between consecutive occupiers of the land - so you should be able to utilise the period of time of the previous owner to make up the minimum of 12 years.
The fundamentals of adverse possession are that you have to show you have factual possession of the land (often done by fencing it), that you have the necessary intention to possess the land, that you are possessing it without the owner’s consent, and you have being doing so for at least 12 years.
Based on what you say, it sounds like that you could claim this and the process requires evidence using a Statement of Truth (ST). You would also need an ST from the previous owner. The LR assesses your claim based on the evidence. It takes some time, so might slow your house sale.
If you are claiming a right of access across the parcel of land that forms part of your driveway you need an easement. One of the ways to achieve this is an easement by prescription and the minimum period of time is 20 years continuous use. Periods are additive so you may be able to use the time of the previous owner. There are various requirements to satisfy, that you mustn’t have been using force, you have been doing it without permission, and without secrecy. From what you say, you wouldn’t have a problem demonstrating adequate proof. You make a claim to the LR with an ST about your use of the land as a driveway on foot or by motor vehicle etc. You would also need to provide a similar ST from the previous owner in order to make up the necessary period of 20 years. Because you say the driveway is known to be unregistered, the LR cannot serve notice on any owner that you are claiming the easement (unless you do the donkey work and find evidence of an owner with a deed for the land – and there is probably no such owner]. For that reason LR will not register your right to access over the land unconditionally. What they do is make an entry in your land title as follows: “The registered proprietor claims that the land has the benefit of a right over the area of land shown as [your detail on the plan]. The right claimed is not included in this registration. The claim is supported by [details of the statement of truth].”
It is in this situation that your purchasers may require an indemnity policy, in case of the unlikely event that an owner with a deed shows up later. I think this is unlikely because the registration of the claim at LR is pretty watertight, as if an owner showed up it would merely allow the LR to complete the easement legally.
Either you are misunderstanding what your solicitor has told you or your solicitor doesn’t understand it properly. And finally. do go back to the solicitor who helped you buy this property and ensure that he picks up the tab for getting you into this monumental legal foul-up. There is no excuse.