When she says 'forced to buzz off' - does she think Sainsburys can and will literally force them out, or that the competition will do so?
I don't think a supermarket could have a cafe forced to shut legally, and I'm not sure they would. Maybe that's naive, but I don't know that they'd want the hassle. I suppose if the cafe had any dodgy hygiene issues (I'm sure it doesn't, that's just a for example), then there could be malicious reporting - but if it had, it deserves to shut anyway.
So, if it's down to competition, well, she can fight them. I don't know the cafe, of course, but if it's near a bus station I suspect it'll have its share of regulars (eg people arriving and departing by bus and those cycling clubs), and there'll probably be a certain amount of loyalty. A supermarket cafe is a fairly souless place, designed to make it easy for customers. So this lady needs to capitalise on her place's character and individuality. Does she do home made cakes for example? A really nice display of cakes is the sort of thing that brings folk back time after time.
If it really is a threat, where along the planning process is it all? If she has an objection, she needs to make it, if it's still within time.
Ashbourne looks like a town big enough to support a couple of cafes. (Googling, I find there's a Costa's too). It's in a touristy area, with walkers and cyclists - people who appreciate a decent cuppa and a nice bacon sarnie and cake. I don't see that a supermarket cafe need be a huge rival.
Good luck to her anyway.