The evacuation actually started in May, not June, but as it proceeded and going into June Churchill was still making adjustments such as (according to his WW2 memoirs, at any rate) pushing for French troops to be evacuated too, and who would form the rearguard. If you want a scenario where everything else until then proceeds, but Churchill is assassinated just before Dunkirk, then you have a date in May, not June. Churchill says in his memoirs (Volume 2, Chapter V, 1951 reprint edition) that '...ever since the 20th the gathering of shipping and small craft had been proceeding under the control of Admiral Ramsay...' with 'the 20th' being the 20th of May, 1940, if I understand correctly. So there's a timeframe; if you want a scenario with just before that starts, that would seem to me to be the 19th of May, 1940.
Communist or Nazi matters, because it seems to me that in a Communist scenario it may harden or alter entirely the British attitude (besides in removing the pragmatic Churchill who said he would give the devil a favourable mention, if Hitler invaded Hell) to the Russians if and when Hitler invades Russia; whereas to my mind, a Nazi or Nazi-supporter does not risk affecting the British mood (except in terms of Churchill's being absent) if and when later Hitler invades Russia.