This is pretty much what Britain was trying to do with Imperial Preference before and after the war. The main reasons it didn't go through were because:
1) The United States was opposed to any deal where the British Empire discriminated against the US. The US wanted free trade with those countries and was an important trading partner with multiple countries there.
2) London lost more and more of its influence among Commonwealth countries during the course of the war. Canada always had to balance its relationship with the US, and Australia decisively move towards the US during the war since Britain proved incapable of protecting it. Even without US pressure, these countries might not think it made best sense to tie themselves so closely with Britain when other associations might be better for them.
3) If by Economic Community you mean the type of economic coordination found in the early European Coal & Steel Community or European Economic Community, that goes against the general desire for Britain and these other countries to do their own thing and not have their sovereignty infringed by multilateral deals.
4) Britain was always trying to balance their desire for closer integration within the Empire/Commonwealth and cooperation with Europe. By going an EC route within the Commonwealth, they are decisively turning their back on Europe and possibly losing influence with France, the Low Countries, Scandinavia, and other countries.
So in general, Britain would lose some influence in Europe and possibly offend the United States at a time when Britain was very dependent on US assistance and cooperation. Immediately after the war, it might give an additional boost to Britain, but long term a pact with Canada, Australia, and New Zealand does not make economic sense by costing you France, Germany, Italy, Holland, and other European countries.
The right time for closer ties within the Empire would have been during the 1920s, or with a very different WWII that did not cost the British so much.