The first question is who does the fence actually belong to?
If you don't have it already then you can download a copy of your title register and title plan from the gov dot uk website. It costs £7 each for the title register and the plan:
https://www.gov.uk/search-property-information-land-registry
Either the title register or the plan (or both) will usually indicate which fence/s you are responsible for maintaining. Although this doesn't always happen.
If the plan does indicate that a fence is your responsibility then it is your fence and you can replace it.
Fences can also be joint or shared fences (in that you are both responsible for the fence jointly). In this situation you are both responsible for maintaining the fence. You can ask the other party to help pay towards repairs but they do not have to, although you can repair and upgrade the fence yourself.
So, check your title deeds and plan and if either
a) you are responsible for the fence, or
b) the fence is a shared fence
then you can replace that fence at your own cost.
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"Landlord allegedly can't afford to replace fence/ boundary (we have offered to pay half for materials and do the labour)"
If it is either your fence or a shared fence then you can replace the fence by yourself but you cannot force them to pay anything towards it.
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If that fence is shown on the deeds as being your neighbour's responsibility then it's a different matter. The fence belongs to them and you are not really supposed to touch it at all.
However, in the situation you mention, I think that there would be a very good chance that if you were to say to the neighbour, "Look, I would like to replace the existing fence between us with a better one. I will pay for it and get it installed" that your neighbour would likely be quite happy with that result regardless of who was actually responsible for the fence.