For example, I know that Australia developed two modernised airplanes: the Boomerang and the Wirraway. However, due to several factors (like a severe shortage of 20 mm Hispano cannons), they were manufactured at a small scale, and therefore saw limited action. We're the efforts of Australian, Indian or Canadian manufacturers significant to the outcome of the war?
If possible, I'd also like to know if other planes or tanks, of British design were "outsourced" to factories overseas, later to be transported and used in the European or African theatre.
Canada built significant amounts of stuff, but it can be a little hard sometimes to separate it from US production.
Canadian shipyards turned out a large number of small craft (mostly landing vessels) but also contributed significantly in terms of merchant production and armed escort ships. The Canadian War Museum keeps an interesting collection of old newspaper articles from the war including this one from The Globe and Mail from 1945 claiming Canada built 5 million tons worth of "war shipping".
The ability of the Royal Canadian Navy to takeover anti-submarine duties in the north-west Atlantic was in large part due to the ability of Canadian shipyards to turn out over a hundred "Flower Class" corvettes (along with a large number of other armed escort craft). This was certainly a significant contributing factor to the ability of the allies to "win" the Battle of the Atlantic.
Canada also had significant military vehicle production. Montreal had a number of locomotive works that were turned over the production of armored vehicles. Frequently these were actually modifications of US vehicles. For example when Canada produced hundreds of Valentine tanks Canada produced, they used General Motors engines. Similarly the ~2500 Sexton artillery pieces produced were based on M3 Lee and M4 Sherman hulls.
Similar things happened with airplane production. Canada made the Lancaster B Mark X, with the distinction of the Mark X being the use if significant amounts of American parts (e.g. electronics). De Havilland built about 1100 Mosquito aircraft in Canada using US and Canadian parts. Companies like Boeing and General Motors contributed significant assemblies to the production.