My upstairs neighbor has had a burst pipe and the water has ran down my walls, I don't have insurance what rights have I got? Thanks

I assume you are renting. The building owner's insurance may cover some of the damage so you should put in a claim with their carrier. You have the right to have the landlord repair the damage and make sure there is no mold.

Depends on who and what caused the burst in the pipe.

The landlord is liable to fix the building, unless some actual fault can be shown, you have to cover your own damages to your property, Landlord fixes his building, you replace your personal items damaged, ( what your insurance would have done if you had it)