Hello I am a lawyer with over 15 years experience I will try to help you with this. Ipf the tree is the responsibility of the local authority -and it probably will be if it is growing on a public pavement- then if the roots are damaging your property the authority may be responsible for taking action.
if you give me a moment I will just double check who the right type of expert is.
Which part of Kent do you live in?
I just spoke to a company called Tree Craft who are based in Sevenoaks I described a similar situation to yours (in Tonbridge) and they would be happy to come round and look at the tree ad give you a quote for free if a report was necessary this would be about £200.00 Their number is 0800 151 411. A fully qualified surveyor might be able to give a more definitive report but I very much doubt that they would do it for that price. Also you may be able to get some decent advice front he tree guy verbally when he comes round. I bet al the tree survey companies would be willing to do this. If you want to call Tree Craft give it to next week as they may think it odd getting to very similar calls in a short space o time.
Once you have got your report of it helps send a copy to the council requesting they fix the problem.