Unless there's a covenant in the deeds of your house, requiring you to maintain some sort of fencing between the two properties, there's no obligation upon you to do anything.
If the fence is yours (irrespective of whether it's damaged or not), you've got every right (unless a relevant covenant applies) to remove the fence and leave the boundary between the two properties completely unmarked. If your neighbour objects, you can simply say "OK, if you want a fence or a hedge, go ahead and provide your own". It follows that there's no obligation upon you to provide any sort of fencing while you wait for a hedge to grow.
Chris