Most of the Holocaust had not happened by May of 1940, so Churchill is more likely referring to crimes committed during invasions. The Allies definitely did not fight primarily because of the Holocaust. In May 1940, Churchill and the Allies were aware of concentration camps, which began to be built by 1933 (these were not death camps, in that they did not have gas chambers, but more comparable to Stalin's gulag). Churchill would also have been aware of the Nuremberg laws, Kristallnacht, and the mass violence against Jews in Germany. He would also have been aware of the creation of ghettos to imprison Jews in cities like Lodz and Warsaw, and the orders to wear the yellow star or blue armband. Mass shootings of Jews and others began as soon as the Nazis invaded Poland, however these shootings were disorganized and very small compared to what would come later. Most Holocaust scholars describe the extermination phase of the Holocaust beginning in the summer of 1941 when Einsatzgruppen were deployed along with the regular army during Operation Barbarossa. Gassing began even later, the first use of gas vans came in late 1941, though smaller scale use of gas was done as early as 1939 in Operation T4. The first death camp, Chelmno, was set up in late 1941 and used gas vans, the first stationary gas chambers came in 1942 with the creation of Belzec, Sobibor, and Treblinka.

I would find it most likely in May of 1941 that Churchill was responding to the violence carried out against all civilians, particularly in France and Poland. There were dozens of mass shootings carried out against Poles, particularly the Polish intelligencia in the first months of war. The invasions toppled governments and displaced and killed thousands. The rates of sexual violence, theft and other forms of brutality are likely extraordinarily high, though specific numbers are unknown. The use of Blitzkrieg to indiscriminately kill, and the constant invasions without provocation. He could also simply have been referring to the tyranny of Hitler being a dictator, and violently siezing territory whenever he pleased. He may have been referring to some of the atrocities already committed against Jews, such as Kristallnacht, but it is somewhat unlikely. For the most part, the Allies knew about the Holocaust as it was happening, and could not or would not respond.

https://encyclopedia.ushmm.org/content/en/article/gassing-operations

https://www.jewishvirtuallibrary.org/when-did-the-world-find-out-about-the-holocaust