Yep, I'm afraid you are stuck with Thames Water as water market not de-regulated, so no competition there, unlike gas and electricity.
However, you still might be able to save money on your water bills. Are you on rateable value for charges, not a meter? if so, and if your property is quite big (ie more than 2 bed) and you are living on your own there or have a small family, you could have a meter put in (installation is free). TW should be able to send you a calculator to work out your consumption to compare meter v rateable value charges, or you should be able to find one at www.thameswater.co.uk.
If you already have a meter, have a look at reducing your consumption. Depending on whether you have an old fashioned cistern or a modern one, TW can provide you with a water displacement device free of charge. You can phone for it or order online. the Hippo is designed for old fashioned loo's, the Hog is for modern ones and is basically a bag filled with silica gel. pop it in and it works in the same way as the old-school method of sticking a brick in the cistern. You can save up to a couple of litres per flush and still ensure the nasties get washed away. It might seem like small beans to start with but over time you can save $$$'s.
Also, there's other stuff you can do like not having the tap on while brushing your teeth, having a water butt in the garden to catch the rain and water plants, etc. It doesn't sounds like much, but in the last 2 years I reckon I have managed to save at least couple of hundred quid that way. It all adds up, and btw I should add I know this nerdy stuff cos I used to work for Thames Water :)
good luck in your new home :)