The Royal Navy used open bridges because its captains preferred them - David K Brown describes the RN as being 'unanimous in advocating open bridges'. There were several reasons why this was the case. An open bridge gave maximum situational awareness. Most of the RN's ships of WWII were designed at a time when the main sensor technology used at sea was the human eyeball. An open bridge gave the greatest opportunities for seeing out, in all conditions. The view from a closed bridge would suffer from spray and rain, even with windscreen wipers, as well as from glare at night. There were also no posts or pillars to block the view from anywhere on the bridge. Given that, in the first half of the war, up to 50% of submarine contacts were made with the naked eye alone, it seems like this was justified. The open design also made it easier to judge the weather, especially the wind. The other main advantage of the open bridge was the ease of communication across it. All of the main officers were on the bridge, along with the lookouts and signallers. There was little to interrupt communications between any of these personnel. Finally, there was a belief that cold air helped to prevent sleepiness and fatigue. While this was later proved false, it was commonly held at the time.

The bridges were less exposed than they might seem, especially to the weather. From Hero and Hereward in 1934, the RN introduced a new form of bridge to its destroyer classes. This was repeatedly trialled in wind tunnels to ensure that wind was properly deflected away from the bridge. As a result, the bridge was free from wind even at top speed. To protect against rain and spray, there was a long glass windscreen, fitted with two Kent clear-view screens to ensure constant vision. Older or smaller ships, though, had more exposed bridge positions, that were less comfortable to use for long periods. The RN moved away from open bridges in the post-war period. The increased spread and effectiveness of radar (and to a lesser extent sonar) meant that naked-eye visibility was much less important. Meanwhile, the threat of nuclear weapons led to a desire to keep the crew inside, away from fallout.

I didn't know that I would be interested in knowing this before I saw the question. So thanks for asking it. It was really interesting to learn the answer.