Known makes that work are Asus, Billion, DrayTek, FritzBox, Ubiquiti, all work 'out of the box' and TP-Link.
With the proviso that the TP-Link would want the beta firmware to work correctly. The latest TP models may have the issue solved, best to check with TP-Link support who may also provide a link to the beta firmware for the VR600/900 routers.
The 'hit or miss' functionality is likely down to the IPv6 issue that BT are working on.
Caveat emptor with Ubiquiti. It's a great router and IPv6 works flawlessly, but the GUI is not that mature for IPv6 currently so you need to 'get down and dirty' with the command line to get things running.
EDIT: Note also that it's a router only and so you also need an external modem.
The Asus DSL-AC68U is a superb router that'll do just about all most users would want to do.
Draytek 2860 works perfectly with IPv6 on BT Broadband and you only have to change one setting to switch it on. It's also one of the few routers that is BT SIN 498 approved.