Is it possible/allowed to keep and breed Yaks in the UK? If it is,
where can I get yaks from?

Should you by chance speak German, you can read about one effort to keep yaks.
Unfortuenaqtely the experiment ended and - as the last line says - 
the taste of ysk steak gives us an idea of how beef used to taste.

http://harzer.koeche.harz.de/Archiv/yaks_im_harz.htm

More helpful, however, could be contacting on this site:

http://www.naomicrose.com/ttlb_yaks.php

Yaks do seem rare in the UK, and probably not easy to import, Quarantine, etc.
German zoos seem better stocked, so I will have to leave you with
Naomi as a reference.
These sites can give you some more information, which you probably already know.

Nice beasts :-)

Myoarin

http://vetgate.ac.uk/browse/cabi/753f1c1c8add30fc2531286cea175902.html

http://www.ivis.org/advances/Zhao/lu/chapter_frm.asp?LA=1