May 1, 1939 - Francis Penin in Paris publishes a calculation of the minimum mass of natural uranium for a chain reaction: 40 tons of uranium oxide, possibly reducible to 12 tons with a neutron reflector. In a paper two weeks later he concludes that a slow-neutron chain reaction will require only 5 tons.
December, 1939 - Rudolf Peierls in Birmingham publishes an improved calculation of the critical mass, extending Perrin's results to neutron multiplication not small compared to 1; results are stated as general formulas.
The article presumably did not reach Germany due to the onset of the war, and the calculations led directly to the realization of about 10 kilograms or less of U235 was needed for bombs.
February 29, 1940 - Heisenberg submits to German Army Weapons the second part of his report dealing with slow neutron chain reactions; he rejects graphite as a moderator.
January 20, 1941 - Walther Bothe and Peter Jensen in Heidelberg measure the neutron absorption cross section of graphite and mistakenly conclude that graphite can not be an effective moderator.
August, 1941 - Fritz Houtermans submits to the German Post Office a report including a critical mass formula like Perrin's but without quantitative estimates, and the suggestion that fissionable plutonium will be generated in a reactor. Houtermans' work gains little attention.
1941 was clearly the decisive year of German nuclear research, as they doubled down against graphite with the ultimate ramifications that being known and Houtermans work, which very nearly came close to solving both the critical mass errors made as well as putting them on the path to a plutonium bomb, was ignored. Given the resource/funding issues, it's questionable if both errors being corrected then would yield a bomb in time for usage in WWII; as noted by Carl, the irony of ironies would be for the Germans to finally get a hold of them in April of 1945. So, that means the easiest way would be to go further back. Basic idea for the ATL could go something like this:
May, 1939 - Paul Harteck stumbles upon Perrin's article and is able to figure out critical mass like Peierls essentially did; the information is shared with Heisenberg. The adopting of the neutron reflector also allows for safety procedures to enter into the nascent German program.
September, 1939 - Heisenberg's first report is submitted, and immediately raises the interest of the German ordnance bureau due to the bomb appearing far more likely. The need to consider a moderator leads to them making the funds available to test for such.
February, 1940 - Walther Bothe and Peter Jensen at Heidelberg are able to confirm that ultra pure graphite can be used as a moderator. With this confirmation, intensive funding and research begins.
Summer, 1940 - Fritz Houtermans produces the research needed to make the plutonium design possible.
May, 1941 - German reactor, in the same vein as Chicago pile, comes online.
June, 1944 - First German nuclear bomb.