"German divisions:

1939 - 78
1940 - 189
1941 - 235
1942 - 261
1943 - 327
1944 - 347
1945 - 319"

The jump from 78 to 189 within a year is a bit surprising- but are probably a number factors which could explain it. One factor could be that Germany didn't want to alarm to the Powers in Europe- and building new divisions and that would certainly be something to tip off these nations. Therefore, similar to what they did in general with other things needed for war, they focused on the capacity rather than build lots hardware in form of planes or ships.
In other words if we in the US wanted to have the capacity to train 400K police force per year, we could train 100k, 100k, then train 400k per year- in other words we could plan on making possible to surge [it also functions as a form of "Shock & Awe.] :)

"Obviously, it is possible to train more than one or two division's worth of troops at a time. Do you recall Bush making any caveats to the effect that it would take 3 years before appreciable number of Iraqi security forces become available?"

I don't think Bush wanted to give a timetable, but it seemed like training the Iraqi army and police was something that was a priority from the beginning. I think that they were hoping that by somewhere around 2006 he could be drawing down US forces and as far I know the US military has done a really good job at training the Iraq army, but as far as I know, the State dept was in charge of training Iraqi police- and I don't think organizationally the State Dept had the same capacity, and I think they made more mistakes in general as compare to military.