My long reply got eaten! There is fairly complete data for May 1940 https://www.revolvy.com/main/index.php?s=British%20Expeditionary%20Force%20order%20of%20battle%20(1940)&item_type=topic (albeit focuses on BEF) & for September, https://forum.axishistory.com/viewtopic.php?t=156666 http://usacac.army.mil/CAC2/CGSC/CARL/nafziger/940BIEA.pdf after the returning forces had reorganised and to some extent re-equipped (although shortages remained).
What is patchy is the period of June to July. There were remnant French and Polish units, returning BEF formations, including some (eg 1st Canadian Division) that didn’t see combat, other Empire formations (Australian and NZ forces, later sent to the ME) & Home Forces units in reserve or training. It’s arguable as to whether there were any fresh combat ready divisions, although there were a number of battle ready brigades.
By August, much of the reorganisation was in train, units were being dispatched to the Middle East and there was a feeling that the period of greatest danger had passed.
Some useful information at http://www.niehorster.org/017_britain/__uk.htm https://forum.axishistory.com//viewtopic.php?t=207767 the latter has a lot of detail!