After much hunting around I managed to find a simple formula which is linked to specific ranges, allowing me to add more staff and more sites as required:

B23(Site A) =COUNT(UNIQUE(TOCOL(VSTACK(B3:S3,B10:S10,B17:S17),1)))

B24(Site B) =COUNT(UNIQUE(TOCOL(VSTACK(B4:S4,B11:S11,B18:S18),1)))

B25(Site C) =COUNT(UNIQUE(TOCOL(VSTACK(B5:S5,B12:S12,B19:S19),1)))