"Access is through a side gate behind our garden."
So this is the side gate that they have locked?
Is this a typical Victorian/Edwardian terrace of homes? Terraces back then were deliberately built with passageways to provide rear access for all the homes and there would typically be a gate in the rear fence leading to each garden.
This allowed people to enter the home from the back door and get coal deliveries etc all without having to traipse dirt through the front room.
As a result, some or all the terraces gained rights over and along the passageways due to their prolonged use over and along them for the uses for which they were intended.
These rights continue to exist today….although many people (like your neighbour) seek to take over these passageways and make them unavailable for the other houses.
So asserting your ‘rights’ over and along the passageway is the issue you have, as opposed to worrying about who owns it.
You will have a right to pass and repass over and along the passageway (to get to and from the rear of your property as a result of long-standing rights after a day down the pit etc in the 1900’s ) and the right to take out & bring in bins, building materials etc (due to longstanding rights bring in coal etc).
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"...it’s now legally their property and we have no right of access and they are not willing to share."
That sounds a bit of a dubious claim, it is possible to check to see if that bit of land is registered and, if so, who owns it.
But who owns the land isn't really so important. What matters is that your house had access along that passageway from the date it was built. That will certainly be long enough to establish a right of way.
You neighbour can't simply get rid of a right of way by locking the gate.
Given what you have said about the neighbour I don't expect that they will be amenable to you just asking (or telling) them that you require access.
If you want to regain access to that passageway then you will likely need the help of a solicitor who is experienced in dealing with rights of way.
If you google something like "rights of way solicitor [name of your town]" then it should come up with some local solicitors who have experience in this area.