There is a video of Grade 1 piano on YouTube and the ABRSM website, made by them, that shows exactly what happens during an exam, what the examiner will say, etc., and that might be quite helpful to watch.
I know that I was always very nervous for music type exams (though luckily not academic ones) and having a very clear idea of exactly what was going to happen, where the room was, what to wear, what to say to the person on the desk, what to say to the examiner, etc, helped me get rid of all the 'extra' worries, so that the only worries I had were the playing ones!
Talking about how exciting exams can be, that they are a chance to show what you can do, etc., is good, because you can explain that some of the physical signs (sweating, nervous tummy, shaking, etc) are because of excitement, which gives them a kind of positive spin.
For academic exams, getting used to doing things with a timer can be helpful - there are lots of computer games etc that have a timer that would work. At first, using one that you can just see rather than hear is less stressful, but then later one that you can hear too - ramps up the stress a bit, and lets her practice staying calm. If it's a fairly easy task - e.g., simple adding, even if she's at the level of doing times tables - then she can get her confidence up about ignoring timers and other pressures, and just simply focus on the task. Again, you can gradually get better at it.
Lots of mock performances are good, even better if she can do background type stuff, so playing when you have a friend over for tea, but not necessarily playing as a performance, just that more people are around at the time, and then gradually build it up to performance if needed.
Get her to practice playing through pieces entirely without stopping - one of the biggest things about music exams in particular (as opposed to academic ones) is that it is the performance that matters, not perfection on little details. She will get far more marks if she plays through without stopping, despite small errors, than if she stops to fix them. But if she is used to practicing to fix things (which is obviously necessarily at times too!), then it might end up happening out of habit in the exam.