Had an UKOOA medical in November. They only pricked my finger for blood. The yellow card was checked to ensure vaccinations were up to date. A hearing test is now part of the UKOOA medical as well. For initial employment the employer may insist on a drug test for which blood will need to be drawn.
Bonesplus, maybe they can give you a tranquiliser after completing the rest of the medical before they prick your finger for blood or draw it? Ask the staff, they are usually helpful.