Oh dear.
By law as many have said already you cannot paint, grow or put anything at all on 'their' fence. However the fence they removed may not have been their property. They are also NOT allowed to place their fence one tidge over the boundary line so they may have infringed upon your garden. In which case you can demand they remove it as they have in fact attempted to steal part of your land.
There is a great deal of misunderstanding about which boundary belongs to one side or another. In fact it's all nonsense. The boundary is an invisible line. either party can erect a fence, wall or whatever or not, as they wish either side of it. As long as the structure sits only on their side.

Neighbour disputes are horrible but it looks like you have to tackle this.
Go in strong with a Solicitors letter outlining these points and that you believe the fence they removed belonged to your property and must be reinstated by them at their expense.

Finish it with a softener suggesting that in the interests of a peaceful life for all you simply request they present legal proof that they have not transgressed the boundary and clear ownership of the original fence they removed.

State that you are happy to accept that and will not pursue reinstatement of your original fence if they can show the fence removed belonged to them and the new one is wholly on their side. If they are totally in the right you must erect your trellis independently. However it can ( legally) be right up against their fence anyway as long as it isn't actually attached. Tell them that is what you will do and express the wish that this can be resolved amicably.
Better to come across tough with your rights and knowledge of them right now
Also put them in a position of having to prove they have not done anything wrong and then show yourself to be reasonable and pleasant. Otherwise they'll not bother to check it out and continue to be arrogant and aggressive about it.
I speak from bitter experience here sadly. Slightly different scenario but initial attempts at neighbourly reasoning met much what you did. Better to make them march off to prove their rights then discover they are in the wrong. It worked for me.