The Daring class destroyers represent the last of the "old" classic destroyers in the RN. Designed in 1943/4, ordered 1945, with the first pair laid down towards the end of that year and completed 1952/3 (the Aussie ships were a few years later - ordered 1946 completed 1957-59). Their AS/AA escort equivalents were the Loch/Bay class frigates and the Black Swan class sloops.
When the 8 WW2 vintage Ca class destroyers were modernised in the mid/late 1950s, 4 (the earliest conversions) were given open bridges akin to the Daring class (including the now preserved Cavalier) and the other 4 closed bridges akin to the then new frigate classes (see below).
All the ships with open bridges had provision for them to be covered by canvas awnings for use in the tropics. In Cavalier's case, this seems to have been replaced by something a bit more substantial since she became a museum ship.
For the RN things began to change in the 1945-47 period for 2 reasons.
1. The need for improved command and control facilities, bringing information on surface, air and submarine threats together in a single place (the AIO, equivalent to the US CIC) in these smaller vessels i.e. smaller than cruisers etc which had gained them during WW2.
2. The advent of nuclear weapons and the prospect of blast & fallout rendering open bridges untenable.
Design of a new generation of frigates began in 1945 but was delayed by lack of funds in postwar Britain. Then in 1947 began the design of the conversion of the first WW2 era destroyers to dedicated AS frigates - the Type 15 with its enclosed bridge. That programme saw 23 R/T/U/V/W converted between Nov 1949 & May 1957 plus 2 RCN & 4 RAN ships. Some of the more austere Type 16 conversions (from O/P/T class destroyers) also received enclosed bridges. The conversion programme would have run to other ships (Types 18 & 62) but was terminated early in 1954 in favour of new construction.
At the same time the design of the new generation of AS/AA/AD frigates continued, emerging as the Types 12/14/41/61, ordered from 1951 as a response to the Korean War and commissioning from 1955 and these were all given enclosed bridges.
The Canadian St Laurent class were designed in parallel with the RN Type 12 (Whitby class), using the same machinery and by a British Naval Constructor, Sir Roland Baker on loan to the RCN, but with a more conservative hull shape. "Reportedly they told their RCNC constructor to make the ships distinctively Canadian" (Friedman British Destroyers and Frigates. The Second World War and After). They commissioned around the same time as the new RN frigate designs.
Again when it comes to considering WW2 era radar equipment the bulk of it needs to be noted. An early (1942) RN PPI radar scope was the size of a desk so couldn't just be fitted anywhere. And initially they lacked repeater screens. By early 1944 however bridge repeater units were starting to become available in RN ships adding to the information picture available on the bridge.
As to the benefits of the open bridge, there are stories from WW2 of RN COs lying on the deck of the bridge when under attack, judging when the best moment might be to swing the ship port or starboard to dodge an enemy dive bomber attack. Couldn't do that with a roof overhead!