How much time would it take to mobilize an army during WW2?

I am assuming that your phrasing refers to the general mobilization of the armed forces of a major nation, typically comprising several armies, air forces and fleets.

In World War One all the great nations of Europe had a two week timetable for general mobilization, with the exception of Russia (six weeks) and Great Britain (6 months). Russia required longer because of both greater distances and a much "thinner" rail network. Great Britain required 6 months because much of its manpower would be from overseas (India, South Africa, Australia, New Zealand and Canada). These Dominion armies would initially mobilize locally, then be transported first to Great Britain and then to France. Barbara Tuchman's Guns of August is a great resource for this period.

Note that the ability to mobilize an armed force eight times as large as one's standing army in just two weeks requires decades of preparation and a specified permanent infrastructure:

1. Mandatory military service of 2 years from 18-20;
2. Followed by 8 years of mandatory Reserve service for 4 weeks / year;
3. Followed by another 10 years of mandatory Landwehr service of 2 weeks / year.
4. A regular standing army ~12% as big as the final desired army
5. A country no bigger than 1914 Germany in size, with a very dense rail network and extremely detailed mobilization plans.
6. A Professional General Staff

Note also that your mobilized units can only pop-up at rail terminuses with sufficient rail capacity to handle the arriving men and equipment, and sufficient open land for their bivouac. The available land around Aachen in 1914 was only 1/3 that required for the destined troops, requiring an invasion of neighbouring Belgium on Day 5 rather than Day 15. This eventuality was unknown to everyone except German mobilization planners.

World War Two is more complicated. Looking at France for example, it was much less prepared than 25 years earlier. Although general mobilization was achieved in "days" (ie perhaps a bit faster than in 1914) it was an utter mess.

• No provision had been made to excuse men working in essential industries such as ammunition production. Over the next few months these men had to be identified, removed from their units, and returned home. In the meantime much of the army was bereft of vital supplies and equipment.

• Many of the the reserve units, comprised of older men, had received much less training than in 1914. This training now had to be planned and executed on an emergency basis. Although this was mostly completed by May 1940, it meant that the French armed forces, while ostensibly "mobilized" by late September 1939, were nowhere near combat ready until spring 1940.

Ironically, and contrary to expectation, through May and June 1940 the Class B reserve units of men in their forties and late thirties significantly out-performed the Class A reserve units of men in their late twenties and early thirties. This is thought to be due to the steadying presence of WW1 veterans in the former.

In contrast, for the U.S. mobilization in World War Two took 2.5 years; essentially from Dec. 1941 until June 1944. Having only a tiny regular army and no meaningful reserve in late 1941, the entire operation of raising and training over 11 million men (thank you Jon Custer) had to be constructed from scratch. The naval vessels that its sailors would man against Japan had, for the most part, not even had their keels laid in Dec. 1941.

The only nation that mobilized a steady-state of about 1 million men under arms, I believe, is Canada. Of these the breakdown by service is roughly:

• Army: 55,000 in 1939 to ~730,000 by spring 1944
• Air Force: 3,000 in 1939 to ~260,000 by spring 1944
• Navy: 3200 in 1939 to ~110,000 by spring 1944

This mobilization took nearly 5 years, from summer and fall 1939 to spring 1944.