If you live in a flat, then I am assuming that you own it on a lease, as will your upstairs neighbours. If you have put them on notice that they need to carry out repairs to prevent damage to your flat and they have not done so, the next thing to do would be to contact your freeholder and to ask them to tell the neighbours to pull their finger out.
If they still don't do the works, the freeholder has the authority to force access to the flat and to carry out the works themselves.
Also might be worth talking to them again, pointing out that the damage is getting worse..? Also, put your buildings insurers on notice of a potential claim (or ask your freeholder to do so, if they arrange the insurance).