There are very many law firms touting for this sort of business, sadly.
I would suggest you help the family by advising them in the first instance to contact the Citizens' Advice Bureau and the local Department for Works and Pensions ( the benefits office) for free unbiased advice.
There would be nothing wrong in contacting one of the advertising law firms by telephone for initial advice, but tell the family they must not sign anything until they are very sure of the costs involved.
This one is good enough to give a freephone telephone number:
http://www.thompsons.law.co.uk/workplace-illne sses-and-diseases/workplace-illnesses-diseases .htm