The reason there are two entries is because the first named is the 'parent' establishment, which would have handled admin duties, such as pay. The name in brackets is the name of the ship he actually served on. This arrangement was because smaller ships did not carry the staff to carry out the duties.
Gloucester III - accounting base at Aden 1939 - 40 (the cruiser Gloucester was Flagship in the East Indies at this time)
Nile - accounting base at Alexandria, Egypt; it looked after most of the smaller ships in the Eastern Med during WW2
Ferret - naval base at Londonderry, Northern Ireland 1940 - 47; it served as both a base, plus the 'parent' for a number of ships based there
Excellent-II - name given to an 'accounting base' - originally set up at Portsmouth, later moved to Westcliffe Gardens, Bournemouth (1942-43) to look after Coastal Forces. In 1943 it was renamed Evolution and had offices in Chelsea Court, London as well as Bournemouth. ....this doesn't mean that your Dad was actually ever in Bournemouth or Chelsea, it just indicates the location of the offices that were responsible for pay and admin etc
The others you mention:
Byrsa - name given to RN establishment at Bougie, Algeria in 1943; later transferred to Castellamare, Italy
Hannibal - name of accounting base established at Algiers in 1943; it had outposts at Oran, Bone and Taranto
Mercury - the Communications School at Leydene, near Petersfield, Hampshire; it was where Telegraphists and Signalmen were trained
Bradford - was one of the ex-US destroyers transferred to the RN under the Lend/Lease scheme.
See:http://www.naval-history.net/xGM-Chrono-11US-HMS_Bradford.htm