One reason comparatively few schools offer scholarships to such young children is that children can plateau as they get older. I know two children who seemed exceptionally bright as toddlers and during infants, and who are still doing well, but not at an exceptional level any more. Also, some of the brightest children don't get going until they are 7, so the competition will be totally different in a few years time. At 5, it can still be difficult to see which child is going to be the high-flyer in secondary school, which is what the scholarship people will be thinking about.
Not saying this to put you off trying to do the best for your dd now, just to be aware that there may not be quite so much on offer now as there will be in a few years time.
Since she is such a young child, I would also consider practical questions about sending her private. Will it mean long travelling hours?- and if so, will that tire her out. Will she still have friends that she can see after school?
Also, it's a good idea to have a fall-back plan, in case she does plateau. I have come across cases where the whole family has suffered because the child failed to come up to early expectations.
Remember that if you do decide to keep her at her local school, there is no rule against stimulating her at home. Lots of books, trips to the library, her own science projects, teaching her to cook, maybe a musical instrument, encourage her to write her own stories- you don't have to have a teacher to have an interesting time. I remember my childhood as a stream of constant exciting discoveries- and I went to the local comp, with no streaming and certainly no provision for G&T people (which I undoubtedly was).