According to the Government everyone who wishes to have a meter installed is entitled to have one fitted, within 3 months of application, free of charge.
I suggest you ask Severn Trent Water to install a meter forthwith and wait to see what happens. They have to carry out a survey to see if the fitting is feasible and that will show the existence of the meter they currently deny. Meanwhile send them (and your solicitor a copy) a letter stating the meter reading on the date you became responsible for payment of the bills. You might also like to ask them how they have been billing the person from whom you bought the flat .
One way or another you arse will be covered.
