Your line mostly decides your terminal
Assignments shift, but the patterns hold. Royal Caribbean and Celebrity sail mainly from the City Cruise Terminal at Berth 101 in the Western Docks, which Royal Caribbean paid to expand. Princess and P&O often use the Mayflower Terminal at Berth 108, also in the Western Docks. Cunard and P&O are the usual tenants of the Ocean Terminal at Berth 46, and the QEII Terminal at Berths 38-39 takes a mix of lines.
The newest, Horizon, opened in summer 2021 behind Dock Gate 10 and was built for MSC and Norwegian. It has shore power, so ships plugged in there can shut down their engines alongside. The five terminals are spread across the docks, so give a taxi driver the terminal name, not just "the cruise port".
Heathrow is 65 miles out, with no train
London Heathrow, the usual long-haul gateway, is about 65 miles from the docks. There is a direct coach but no direct train, so the choices are coach, a pre-booked car, or the cruise line's own transfer. Gatwick is farther, around 95 miles, but does have a direct train to Southampton.
From London itself, trains to Southampton Central take a minimum of an hour, then a short taxi to the terminal. Stacking an overnight transatlantic flight, a 65-mile road transfer and check-in onto one day leaves little room for a delay, which is why many passengers arrive the night before.
Sources: beyondships3.com, cruisemapper.com