As to the UK manpower issue...I feel 18 divisions is greatly underestimating the available manpower. While there probably wont be as many available as in WW2 (around 70), there certainly will be more than 18.
There is also a LOT of manpower available in the Empire (quite a lot of troops were held locally in WW2, for example in Africa. With no credible threat outside Europe, some of these will be available). Indeed, its almost certain that for political reasons the invading army will comprise troops from as many countries as possible.
Remember also that the need to keep large standing armed forces during the intervening years will drive efficiency in industry even more strongly than in OTL, and once the German navy is admiring the bottom of the Atlantic the manpower in the RN and USN can be reduced somewhat.
18 was the estimate for the field army size deployed to NW Europe.
In the historic Second World War, Britain deployed around 5 million men in total at its high point in 1945.
2,900,000-3,000,000 in the Army
865,000 in the RN
1,100,000 in the RAF
At the close of the war, this translated to 3 Armoured Divisions (plus 79th broken up around the place), 2 Airborne Divisions 18 Infantry Divisions and 4 reserve/holding divisions in the UK that weren't going anywhere. There was also quite a lot of manpower tied up in Anti Aircraft Command.
We'll even back track and change that to 4 Armoured Divisions, given that the 1st was disbanded in early 1945. That still gives only 24 deployable divisions, plus a lot of independent artillery regiments, AGRAs, armoured brigades and copious support units.
3 African divisions were in service, along with the Caribbean Regiment.
Extra manpower will be available from the British units no longer serving with the British Indian Army, which will balance out losses (such as the airborne division over the Channel Islands) and the presumed loss of all POWs captured by the Japanese (the starvation approach taken will have meant the death of virtually all Allied POWs, plus millions of Chinese and Koreans).
Strip the garrisons around the Empire, as was done at the beginning of WW1, and perhaps 2 divisions would be available.
Efficiency is not something that will occur overnight, and doesn't affect all of the big manpower industries - coal mining being an important example. Efficiency in and of itself will not give more than 1-2 divisions for the Army, when requirements are spread out across the military.
The strength of the RN may fall, but a lot is needed for supporting the invasions, including heavy manpower ships such as battleships, cruisers and carriers, not to mention a lot of landing craft, logistical support ships and transports. Let us work with that though, and reform the Royal Marine Division and a Royal Naval Division (although the latter may not be called that in this situation).
Forces will still be required in Malta (2-3 brigades), Gibraltar, Cyprus (2-3 brigades), Egypt (1-2 divisions), Palestine (1-2 divisions) Iraq (1-2 divisions) and Persia (1-2 divisions), not to mention Malaya and Hong Kong (1 division between them). These will have to be fielded without recourse to the British Indian Army, as India is basically independent in this scenario and we perhaps cannot count on them garrisoning British Imperial possessions out of the goodness of their hearts.
Some of those locations will need a strong garrison as they hold very important airbases for the strategic bombing campaign, such as the Middle East.
Even with a larger population of 51 million as compared to 48, Britain is not going to have a huge amount more than 5.5 million men under arms, as that is a very large slice of the available industrial workforce, which is around 19-21 million.
That would potentially give us
Airborne: 2 divisions
Home Defence/Training/Holding: 4 divisions
Strategic Reserve: 2 divisions
Minimum Imperial Defence: 6 divisions
British Contribution to Allied Expeditionary Force:
1 Army
4 Corps
14-18 infantry divisions
4-6 armoured divisions
+ independent armoured brigades, infantry brigades, AGRA, artillery regiments, engineers, logistical support, reconnaissance regiments, lines of communication troops and light/medium AA regiments
Add to the pot the African Corps - call it the British African Army if you want to.
So, around 32-40 divisions all up, but only 18-24 that can realistically be deployed to Europe.
That force will need to be shepherded carefully, as, like 21st Army Group, it is Britain's last field army and cannot be replaced. It will probably see divisions disbanded to cater for losses based on the course of the campaign thus far.
This is not taking into account the huge cost of maintaining a massive force of V-Bombers, and then using them as jet powered Lancasters in a conventional strategic bombing campaign, taking appalling losses in blood and treasure.
The cultural effects of what type of campaign is fought will be significant.
If the British Army manages to punch above its weight and win several battles against the Nazis, then the future image of the war in popular culture will be tinged with evocations of great heroism and triumph, although at great cost. If they get mauled, then it will be a bit more like the First World War, where the losses are an area of focus.
Both of these will be decidedly secondary characteristics to the characterization of the war as a crusade and a just war against an unspeakable evil; but the butchers bill will affect the lighting, so to speak.
The culture that grows out of such a war, and out of half a decade more of hard austerity, will be an interesting one. I can see absurdism and escapism coming to the fore to some extent, but there may be a need for a strong army of occupation. This would delay the sense of collapse that lead to a lot of the anti-establishment comics and Angry Young Men playwrights that emerged from the mid-late 50s; historically, it came from the loss of Empire, and therefore the loss of a lot of career paths into the Colonial Office for many educated young men.
The damage done by German rocket attacks and air raids will result, sadly for some, in the wider spread of brutalist architecture in the postwar rebuilding phase.
National Service is unlikely to end any time soon, so the potential building material for a lot of the musical groups of the late 50s and early 60s will not be at a loose end or at art school, but will be out across the world with a rifle and short hair cut, down the coal mines or on occupation duty in some areas of Europe.
This is pure speculation, as it depends on
1.) How the war goes
2.) Any potential Himmlerian Gotterdamerung/use of chemical and biological weapons in the death throes of the Nazi regime
3.) Postwar US policy
4.) How, when and where atomic weapons have been used
5.) Whether Britain collapses economically
6.) The postwar economic order
It is a panoply of bad situations and unenviable choices, all of which will cost huge amounts in blood and treasure. This would lead to increased pressure for a much cheaper, quicker and more decisive means of destroying Nazi Germany, in the concentrated use of atomic weapons, or anthrax or anything that can have a massive and immediate effect.
If they (atomic bombs) are regarded as simply another weapon (or, to use an apt phrase one once heard, 'a very big bomb with some interesting side effects' ) then there is a case for using them in this situation. If they are not used, the question will be asked as to why so much money was spent building them in the first place - a question that will always come up in World War 3 scenarios or world wars that take place in the atomic age.
Even though we have the oft-cited case of Japanese starvation and devastation, this is much, much closer to home from a British perspective. British cities are getting smashed and another generation is being bled dry in France. The Empire has been lost, and Britain is on the precipice of economic collapse, and there is precious little to be shown for it.
There will be no question that fighting and destroying Nazism is the right thing to do. There will be a lot of questions, at the appropriate levels, as to why it is being done without using the most powerful weapons available.
