The good thing about being in Great Britain is that places aren't particularly far apart from one another, compared to the situation in America. Therefore, it is probably likely that you would be able to see pretty much anywhere – certainly in England – that you would like to see, without having to worry too much about transport times.
That said, my suggestion would be that you would be best off starting in London – this is probably where you would fly to anyway, and whilst the zoo has its critics it is still a great place, certainly worthy of a day's visit. Nearby are Colchester and Whipsnade – each worthy of a day. To the south east of London, in the beautiful county of Kent, can be found Howletts. Many of us have issues with the place, and I must admit that I would not choose to visit it for many reasons, but nonetheless it should be seen; if you are here at the right time of the year a great doubleheader could be provided by the nearby Rare Species Conservation Centre. This is due to reopen in the spring of next year.
A slightly longer journey from London will take you to the north-west, and Chester. Nonetheless, this is still a very do-able drive – just a few hours. Although it is certainly not perfect, Chester is probably the best that England has.
About an hour and a half west of London, you will find the Cotswold wildlife Park – rated by many as one of the nicest in England. It certainly is very beautiful, and worth visiting. Other alternatives would be places such as Twycross and Marwell. Neither is far from London, and each has its merits – although their respective shortcomings are heavily chronicled on this site.
It is for sure that if you come to England, while you will see many zoos, you will not see the quality to which you are used in North America, nor will you see zoos as good as those to be found in Holland, Germany, the Czech Republic, Switzerland or France.