Looking around others have replaced the HG533 and are saying just plugging into the Ethernet WAN port of their new router from the Openreach modem works.
As for which router, the two fast ones I've used and am happy with are
Asus RT-N66U - more expensive but great to use and nice web interface
The TP Link TL-WDR4300 also pretty quick, but a web interface more suited to the geek than the casual home user.
I would be slightly wary of the really cheap Ethernet routers, unless you find someone using it on FTTC (Fibre) and are getting the best speeds from it. i.e. cheaper routers tend to be slower processor and less memory.