I think after 5 days she can pay to do another test to potentially end home quarantine early, doing the 'Test to release' option.
Check this link: https://www.gov.uk/guidance/coronavirus-covid-19-test-to-release-for-international-travel
As for getting vaccinated in the UK - it should be no problem at all. Check this out: https://www.nhs.uk/nhs-services/visiting-or-moving-to-england/visitors-from-outside-the-european-economic-area-eea/
But to be fully vaccinated she will need to obviously remain in UK until she get the 2nd dose (time between doses will depend on vaccine). And