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 then wait 14 days after 2nd dose before being considered fully vaccinated.
Good luck