As far as what the leaders would be would depend on not so much what the structure of the Parliament is, but what the objective is. If you're oging to make any sort of Imperial Parliament work, you must - I repeat, MUST - make it very worthwhile for the Dominions to be willing to give up ultimate control over their countries, and what is needed to be given up? Foreign policy, trade laws and military leadership are obvious, but beyond that there is a number of yawning differences between the needs and desires of the Dominions and that of Britain itself.
If this is done in the late 19th Century or very early 20th, the problem of Ireland's ultimate fate probably lands in the Imperial Parliament's lap, which is a serious problem from a political standpoint as approving of Ireland's breaking away from the Empire will cause you to have the same problem with the Afrikaners (a certainty) and French-Canadians (quite possible). Knowing that, perhaps the end result is that Ireland is allowed to become a Dominion as a compromise? Likewise, New Zealand would at first have a real issue with the Maori likely being in line with Native Canadians and colored South Africans. Giving control over foreign policy to the Imperial Parliament also means they have to deal with the issues surrounding India, and as people mention Palestine also would become a problem. If the result is a demand for equality across the Empire, the Imperial Parliament would be immediately facing issues with regards to regressive social treatment across the Empire - Apartheid, White Australia, Canada's treatment of its First Nations....lotsa problems here.
The Caribbean territories would probably eventually be integrated into Canada (except for the Falklands, St. Helena and Bermuda staying colonies), with South Africa gaining South-West Africa, Bechuanaland and Southern Rhodesia and many of the smaller islands around Australia from as far east as Pitcairn to as far west as Christmas Island becoming part of Australia or New Zealand. Malta and Cyprus would be possibilities for parts of the Empire as well, particularly the former. Israel would be a very real possibilities, particularly if the Empire forces a solution that allows both sides access to Jerusalem in some way. If Ireland remains a Dominion (I grant this is unlikely, but not impossible if the Imperial Parliament allows the Dominions wide control over domestic policies), it may well dodge The Troubles in their entirety and if not they would likely be much reduced in size and the country would probably be less intertwined than the UK of IOTL, because being a Dominion and with Irish all over the world they'd have the ability to work with all of the rest of the Empire without the need to involve the UK.
But assuming they got over all of the most serious difficultes....it would be invariable that the nations would use each others' truly-immense natural resources for their own advantage, and that of their closest allies. China would probably never get the chance to be the world's factory because the Commonwealth would make absolutely certain India gets it first, and while India wouldn't likely fit in the Commonwealth, it would absolutely be tied to it in pretty much every way possible. China would probably eventually emerge from the Maoist era, but even if they demanded Hong Kong back, they'd not get it, though they would probably get a major sum and priviledged access to the Empire in return for relinquishing the New Territories. Aden would be the same, and by 2020 Aden and Singapore would be two of the richest places on the world as a result of the massive trading that would be heading from India west to Europe and east to Asia and Australia, and with goods heading from the booming East Asian economies to Europe via Singapore. Canada would be the UK's energy source via a pipeline to Halifax (and later probably supplies from offshore platforms on the Grand Banks) and tankers to the British Isles, while India's massive heavy industry is fueled by minerals from South Africa and vast supplies of Australian iron ore, while food moves to the more-densely populated areas from Canada and Australia in unimaginable quantities. The Empire's inter-war desires for growth of value-added industries would create the genesis of some of the world's largest makers of automobiles, railroad equipment and aircraft, and standardization ultimately comes to some degree - Canada eventually begins driving on the left, railroads are built in standard gauge with buckeye couplers and British-standard signalling, the metric system comes across the Empire in the 1950s and 1960s. After World War II, the ever-better systems of air travel created by British and Canadian-manufactured jet airliners results in the Empire's respective populations moving around quite frequently, bringing with it many of the cultural interests. Canada joins their Empire brothers in the competitions of rugby, cricket and football (and prove shockingly good at the first of those rather quickly....), while India's taking their place as the Empire's industrial ally in the second half of the 20th Century massively increases their country's wealth and cultural influence, even as the connections make things go the other way to a large degree as well, particularly as Indians of wealth begin becoming a common sight across the Empire starting in the 1960s.
Established minimum prices for agricultural goods during the Depression ends up being a massive benefit to South Africa most of all, and as a result of this and the massive resource wealth of South Africa by the 1960s the Afrikaners of South Africa are some of the richest people on Earth, even as land reform and more people moving off of the land results in a titanic growth in income for South Africans of colour after the War. (This also all but eradicates the anti-British sentiments among the Afrikaners as time goes on.)
London is the absolute financial capital, but there are numerous capital centres with massive resources of their own - Toronto, Melbourne, Hong Kong, Singapore, Johannesburg, Mumbai. British banks and financial institutions dominate the Empire early on, but the biggest of the Dominions quickly gain a major presence in the post-war era. The British pound sterling is used as the reserve currency and is the currency the natural resources within the Empire are traded in - something that helps the UK's economy immensely - but the currencies of the individual nations are all strong by themselves, particularly the Canadian and Australian dollars. Banking regulations between the nations are standardized in the 1990s primarily on the Canadian mold - this done because of the Canadians' reputation for excellent regulatory management - and the result is that the City of London is by the 2000s every bit the equal of Wall Street if one wishes to go find money, particularly for places in the world that have a lesser view of Americans.
The Royal Navy and Royal Air Force are the established global armed forces for the Empire and all ships carry the HMS Prefix, with military leadership being the Imperial Parliament's job, with the Admiralty and the Air Forces leadership being based in the UK, but with explicit divisions established. The Royal Navy is divided into the Home Fleet, Atlantic Fleet, Mediterranean Fleet, Indian Fleet, Asiatic Fleet and Pacific Fleet, each with their own established bases and command centers, likewise the Royal Air Force, which is divided into First (UK), Second (Australian/New Zealander), Third (Canadian), Fourth (South African) and Fifth (Asian) Air Forces. The Royal Marines are similarly divided into Brigades that have their own established bases across the world. The Armies of the Empire are a little different as there are British, Canadian, Australian and South African Armies and Armed Forces of New Zealand, Ireland and all of the smaller Dominions.
The abilities the combined forces have is absolutely immense - compared to the American armed forces they have a rather smaller total army (though still huge) and a smaller carrier fleet, but a massive Expeditionary Fleet to support the multiple Royal Marines brigades, a very large submarine force (both nuclear and advanced conventional), a massive surface fleet and logistic assets to support all of the above, as well as a gigantic air force with worldwide reach and immense logistical and airlift capacity. Virtually all of the equipment of the Empire's armed forces is made in the Empire itself, and the Empire's armed forces are trained very well indeed, with training in many different environments to suit individual missions - cold weather in northern Canada and hot weather in the interior of Australia, for example.
Politically, the belief of the growth of the Empire as a force for good and for equality of peoples defined its politics throughout the post-War era, particularly as it was felt (not unreasonably) that without the Empire being for all who lived in it it would invariably come apart, the horrors of World War II and the nastiness of Indian partition being hoisted as a real to make sure all who lived in the Empire benefited from it. This focus on the balances between the rule of law, social and economic equality, resolute honesty and as great a personal freedoms as is possible is the driving force for the Empire in modern times, beginning with the immense efforts to improve the lives of blacks of the Caribbean, Native Australians and Canadians, the Maori and all of the various minorities of South Africa in the Post-War era being the genesis of many policies. The Empire of 2020 has long ago condemned bigotry as totally unacceptable in politics, and since the 1980s has pushed new frontiers in the steady growth of LGBTQ rights, which by 2020 are also nearly universal. Individual Dominions have long held differing viewpoints on what is acceptable in society, but as travel and communications improvements have brought the Empire closer together, many of these viewpoints have closed up.
The political parties - Conservative, Labour and Liberal are the three largest ones, and there are many smaller ones, with this diversity of representation existing to such a degree that a single party having a majority of the Imperial Parliament is regarded as almost an impossibility - tend to be led by strong, capable leaders, as they have many good candidates to choose from. Being a Member of the Imperial Parliament is a highly-prestigious position in any right and so the parties are very capable of finding highly-qualified candidates for both party positions and cabinet positions, resulting in a "Imperial political class" that is almost entirely drawn from highly-qualified men and women, an environment that has encouraged the parties to look out for who might be the next great leader. Originally the leaders tended to come from British backgrounds, but starting with the Depression-era years and particularly during the Imperial Prime Ministerships of Winston Churchill (1934-1942), Jan Smuts (1942-1946) and Clement Attlee (1946-1951), the best from the Empire began entering the Cabinets of the Imperial Parliament, a tradition that grew far beyond the first Dominion-born Prime Minister (Smuts) to include a great many of the best from the Empire. Conservative successor, Anthony Eden (1950-1955), was followed by Canadian Lester Pearson (1955-1964), who was then followed by the first Australian Imperial Prime Minister, Gough Whitlam, who assumed the post in May 1964.
Imperial Prime Ministers
Joseph Chamberlain - 1902-1906
Henry Campbell-Bannerman - 1906-1908
Herbert Henry Asquith - 1908-1916
David Lloyd George - 1916-1924
Stanley Baldwin - 1924-1929
John Robert Clynes - 1929-1934
Winston Churchill - 1934-1942[1]
Jan Christiaan Smuts - 1942-1946
Clement Attlee - 1946-1950
Anthony Eden - 1950-1955
Lester Pearson - 1955-1964
Gough Whitlam - 1964-1970
Tommy Douglas - 1970-1974
Robert Stanfield - 1974-1982[2]
Edward Heath - 1982-1986
Robert Hawke - 1986-1991
Nelson Mandela - 1991-1998[3]
Edward Broadbent - 1998-1999
Tony Blair - 1999-2008
David Cameron - 2008-2014
Justin Trudeau - 2014-2018
Jacinda Ardern - 2018-present
[1] Resigned due to health issues, remained in War Cabinet under Smuts, remained as a member of the Imperial Parliament until 1962
[2] Retired in an attempt to not bog his party down during an election, Heath's victory seven weeks later was proof this plan worked
[3] Retired at Age 80
