Trading Standards is the nearest thing we have, but they have serious legal teeth :-)
But there is also the Citizens Advice Bureaux, who will deal with some of the things that in the USA are BBB, as well as being all round good guys when it comes to advice of all kinds (legal, financial difficulties, access to state benefits etc).
When it comes to tradesmen, you can contact the various trade bodies to 'check up' on their members; and there are various websites that list reviews of people (although a couple have issues atm, as they don't seem to be allowing negative reviews to stay up!).
It really depends on what you want as to where you would go.