Regardless of the legal technicalities, the sooner you advise the employer that you won't be joining them, the better.
I don't know how long the notice period in the contract you signed is, but with a month's notice they'll be annoyed but they'll at least have a chance to readvertise the job. They'd have been foolish to promise you to clients as a resource on any contracts, but in case they have they'll at least have time to get a contractor to cover any immediate work.
They are very unlikely to sue as it serves no constructive purpose - unless you are a critical executive and on a substantial pay packet, there is no commercial advantage to be had.
[I am not a lawyer: this does not constitute legal advice]