I'm sure I don't have a definitive list but I'll mention those I know of..:
And best of all (cos they have me) , Dawn of Chivalry.
I think most focus mostly on 12th/early 13th, we focus on mid-13th.
Oh, and there's a whole bunch of hostpitaller groups, many doing 3rd crusade, and I think Regia are moving into that period too. I'm sure there are a lot more that I don't know of or have forgotten about. Hopefully others will add them.
Shameless plug time:
Dawn of Chivalry are based in the North-East, covering the mid-13c (but can also do 12c), focusing on delivering a small, but pretty darn good foot tournament. We have a full period camp and this year will be providing the English Heritage foot tournament in many of their northern castles (including Richmond, Carlisle, Belsay). And also travelling a bit further afield, to do the battle of Evesham and History Live.
Oh, and did I mention, they have me?
We have the events, the infrastructure, provide training and equipment, just looking for a few more willing souls.