Checking your property title deeds should be the first port of call.
Checking those of your neighbours too, if the boundary feature divides your respective properties. If their property is registered at the Land Registry, this is as simple as entering an address on the Land Registry website.
The wording of the deeds may indicate the position, or typically, if your house has been built on a new estate, then it is likely that there will be a scale plan showing the garden boundaries by reference to 'T' marks on the plan, which usually refers to who has responsibility for the maintenance and repair of a boundary feature.
However, this is not conclusive, as fences blow down, new physical features can be erected with neighbours clubbing together. But checking the deeds remains the first port of all.
On private land it is the landowner's responsibility to fence their livestock in (Animals Act 1971). However, this is not the same as proof that the fence is the landowners, but it is a starting point. (Otherwise, there is no such duty on a neighbour to fence their land from yours, and if they want to allow any current fence to fall into disrepair, they can, so long as it does not fall on to your land.)
In the absence of 'T' marks or words confirming the position in the deeds, a wall is presumed to be jointly owned and maintained.
Unfortunately, property deeds are so frequently silent on the ownership or maintenance of boundaries. Failing anything mentioned in the property deeds, there is a rebuttable presumption that a person will have chosen to erect a fence and therefore own it, if they place the 'ugly side' - the posts and arris (horizontal) rails - of a fence on the owner's side.
Where a fence or hedge has an adjacent ditch, the ownership boundary is the edge of the ditch furthest from the fence or hedge. The presumption being that an owner digging a boundary ditch will normally dig it up to the very edge of their land, and will pile the soil on to their own side of the ditch to avoid trespassing on their neighbour's land. They may then erect a fence or hedge on the spoil, leaving the ditch on its far side.
There are lots of different beliefs that the way a wall or fence is constructed can suggest who the owner is, for example ownership is always on the right or left when you look from the front of a property. However, there is no legal significance to this.
If the physical feature needs repair/maintenance, then ask the neighbouring landowner, and if they claim ownership, and you are content with that, then have them confirm this in writing.