There are several local doctor's surgeries:
Carterton (part of the Burford Surgery Practice) ~
http://broadshireshealthcentre.com/ and http://www.bamptonmedicalpractice.co.uk/
https://www.nhs.uk/services/gp-surgery/carterton-health-centre/K84047001
(at the Carterton Health Centre on Alvescot Road)
Lechlade ~
https://lechlademedicalcentre.co.uk/
And there is a Minor Injury Unit in Witney at Witney Community Hospital Welch Way, Witney OX28 6JJ, 01993 209458. It is listed under the Oxford NHS website at https://www.oxfordhealth.nhs.uk/service_description/minor-injuries-units/