My wife is Brazilian, and she is craving yams. Cannot buy them anywhere (can order online but costs stupid money).

Has anyone here successfully grown them in England? If so, how?

There are several different plants known as "yam", not just the quite various Dioscorea species, but also some other quite unrelated tubers such as sweet potato and konjac (taro). Check out http://en.wikipedia.org/wiki/Yam_(vegetable) and try to work out exactly what it is you want to grow.

Things called yams are sold in Brixton market, but they may not be the ones you have in mind. Various things which have been labelled yams have from time to time appeared on the shelves of my local supermarket (Waitrose), but again may not be the ones you have in mind. Presumably it isn't sweet potato you have in mind because that is quite easy to buy here.

No doubt you can grow whatever it is you want to grow if you recreate the right conditions in a greenhouse, but it may not be easy or cheap to do that. If its konjac (taro) you want, then people grow that quite easily, or at least certain types of it, merely needing to keep it frost free over winter. But I suspect the Dioscorea species are more difficult to grow, especially the more equatorial ones.