I don't know if any of these places are near you, LB, but they offer a walk-in service.
//Eligible adults can also walk in to one of our mobile vaccination sessions.
Weekly sessions are available:
•Wednesdays: 10.30am to 2.30pm, Tesco superstore Hove
•Thursdays: 10.30am to 2.30pm, Outside Hove Polyclinic
•Fridays: 10.30am to 2.30pm, Wellsbourne CIC, Whitehawk
Other walk-in sessions are also available. Please check the Sussex Health and Care website for the latest information.//
https://www.brighton-hove.gov.uk/coronavirus-covid-19/vaccinations