Thanks for all your feedback, it's been very useful.
I think this means that I am going to be better off trying to get a professional drain company to try to clear it because the risk of Thames Water deciding just to dig the road up and charge me for several thousand pounds is too horrendous to contemplate.
Yes but surely Thames Water would inform you in advance of the work should they decide to charge for the job.
I'm sure they would if it were our drains and we were requesting the work, but because the blockage has been pushed down into the part of the drain that's owned by them, once they know it's blocked I believe they can do what they like (even though it's only the drain from our house that's obstructed). My understanding is that once they choose their course of action, irrespective of our wishes/capability to pay, they can then pursue us retrospectively for the cost of whatever they did.