Find size of incoming water main

Mains water pipes are usually found in the road outside your house and a utilities search conducted by one of many companies will show this along with the pipe diameter. If you are referring to the pipe between the main and your water meter or outside stopcock, if Thames Water don't know this then the only hard and fast way is for the correct party to dig up the road and have a look! It may be an old iron main that has corroded and narrowed. I would do a lot of investigating before committing £6k, such as chat with your neighbours to see if they have a similar problem / what their setup is like and check that your stopcocks are not partially closed. If the water pressure is low, say 1 bar, you are always going to struggle. If you have a direct system, check the pressure limiting value to see if the gauze is clogged up. If everything points to the narrowness of your incoming main then coughing up to Thames Water may be the solution. £6k sounds high but they may have a long pipe run.

best to get a decent plumber to check other options first, could just be a slight blockage in a pipe, usually at a fitting like stopcock,would just be case of turning off water on street, disconnect stopcock in property and go from there also if issue was with supply coming to your home and not a blockage then it has to be supplied at a decent pressure and would be down to water supplier / council to fix, again a plumber could check your incoming pressure and advise you from there

You could all ways tee into the incoming supply after the water meter into the property & measure the flow rate & pressure. If found to be adequate then no use paying 6k as the problem is on your property side. A average mains from the meter into the house would should be a lot less than 6k

maybe a salamander mains water booster would solve your issue or mains water accumulator vessels .. much less than 6k and Mayb enough to satisfy demand