I am labouring this point because having the right of way noted on the Land Register as a matter of fact is the optimal starting point if you seek to clear the overgrown strip of land which you say comprises the right of way, and exercise the right.
A clear impediment to your right of way is the garden office, which you say has been built across part of it. Whoever owns that building may argue that the fact it is there, and has been for some time, equates to the previous owners of your cottage having abandoned the right of way you are now seeking to enforce.
You use the word “reinstate”, but I would avoid that phrase as it infers the right has been lost, when your legal argument is that it has been there all along, albeit exercised infrequently.
There is some weight to an argument that a right of way that is blocked has been abandoned. Further, the legal “doctrine of laches” states a person with a legal right has a duty to assert that right or risk losing it.
However, in law, it is hard to prove an express right of way has been abandoned. Your rebuttal to such an argument would be that, although the path of the right in question has become overgrown and has narrowed, it is still there and is an identifiable route not completely blocked by the garden office.
In terms of what to do next, I would get your legal paperwork in order as outlined above, and then either approach the owner of the strip of land and say you wish to exercise the right of way. Or just cut back the overgrown path and start using it.
It is not clear from your question if the owner of the path is the same as the owner of the garden office. But if they are the same person, your leverage is that the building is infringing your right of way, so you will only accept the building if they allow your right of way.
One final comment is that as you are new to the locality, making some assessment of how important this issue is to you compared with not upsetting your neighbours would be no bad thing. It’s one thing having legal rights and asserting them. It’s quite another to assert rights with no real-world benefit at the expense of forging close connections with those who you live among.
Ask a Lawyer should not be taken as formal legal advice, but rather as a starting point for readers to undertake their own further research.