Hi Dave and thanks for your email. Hypermobility generally means that you can end up putting your joints on a little more end of range position than others. In terms of typing though this is not going to affect your position in the wrists more than anyone else (as it is a mid range position). Of course the spinal positon, especially the neck alignment, could be affected more in your case.
Wrist issues tend to creep up slowly and only become symptomatic after a long time. This means they are not going to subside quickly. I have found that often this can be as simple as the way the tendons around the wrist end up resting on the edge of a table, causing prolonged compression that can lead to dysfunction. In my case I suffered from this and the only thing that helped was a support used under the wrist (not actually the hand) and of course not working in a way that lets my wrist settle onto the tables edge. Attached is an example of the king of thing I mean. I do also have one mouse pad with a nice support area for the carpus of the hand that works quite well too but having tried several different ones there was only one that felt OK.
Also consider the tension in yourself when you are working on the keyboard. I have had times where I was working in a too stressed environment. This would cause me to grip the mouse a little harder, and even get sore fingers from tapping to hard on the keyboard. In fact if I think about it even more then best thing for me was to not use a mouse at all but to use a trackpad - something that happened by accident when moving to using a mac laptop more than the desktop. You can get trackpads to use instead of a mouse so might also be something worth trying.
In terms of addressing current symptoms you could wear a simple wrist support but I have found a nice wide tennis wristband has the same, if not better effect at spreading the tension/load across the flexor tendons, thereby reducing possible median nerve irritation. Maybe try a few of these ideas, consider it will not be an overnight success but could take a few months to resolve or reduce. So long as things aren't getting worse you are on the right track.
If things don't move in the right direction then get to a specialist hand therapist to assist with a more accurate structural diagnosis and respective treatment plan. Hope this helps you a little and let us know how you get on. :-)