Would Russian occupation of western Europe lead to the U.S. and U.K. going to war with Russia at a time when only the U.S. had nuclear capability? Certainly possible.
I don't think a failure on June 6, 1945 would have led to complete Soviet domination in Europe for several reasons.
First, the US/UK invasion of Italy was still proceeding and the later US/UK invasion of southern France could still have been conducted. Additionally, even if the invasion on June 6 had failed, it still would have been reasonably possible to simply keep stockpiling material in Britain and try again later.
Second, and more important, the US conducted the Trinity Test just a 13 months and 10 days after D-Day. On July 16, 1945 the world entered the nuclear age about 35 miles from Socorro, New Mexico.
As it happened, the Russians got to Berlin in early May, 1945 but Europe's preeminent anti-communist had wasted huge amounts of German war-making capability on a futile effort to turn back the US/UK forces the previous December (See Battle of the Bulge). Without a US/UK army conquering France and threatening Western Germany as well as capturing much of Germany's industrial heartland in the Ruhr it is obvious that the Germans would have been better able to delay the Soviet advance. I'm not saying that they could have stopped the Russians, just that they could have slowed them down more than they already did.
What this means is that even if D-Day had been a colossal failure, there still would have been viable German targets for US atomic weapons when those became available in August, 1945. The net result would have been a MASSIVE increase in German civilian casualties as US Atomic weapons obliterated German Cities. B29's operating out of southern England could have reached any city in Germany.
There is also another scary possibility. In preparing to invade Japan, one possibility that was considered was to use a/an Atomic Bomb(s) to clear the beaches immediately before landing. The long-term consequences for the US/UK troops landing on recently nuked Japanese beaches would have been catastrophic but that wasn't well understood then and the Army had a war to win. If D-Day had failed it is not unlikely that A-Bombs would have been used to clear the way for a second try in August, 1945 with the Russians probably still in Poland.
I don't think that Soviet domination of Europe would have been total largely because from July 16, 1945 until August 29, 1949 the US had an absolute monopoly on nuclear weapons. Even after that, the US had much better delivery systems than the Soviets for another decade or so.