Here's a few numbers, and a couple of thoughts. I come from a family with a rather long military background.
The information is a matter of record and easy enough to find.
There were quite a few Muslim troops in WW2.
You could start with the Senegalese Tirailleurs that fought in the Battle of France in 1939, later as part of the Colonial Infantry Division within the French First Army.
Who's to say the Indian Army (remember Monte Cassino and the Winter line, and the subsequent Gustav and Gothic lines?) didn't ease the taking of France, in conjunction with troops from the US, Britain, New Zealand and the French, because those German troops, and later reinforcements, couldn't be redirected to Normandy?
(The Allied force involved at Monte Cassino, the US Fifth Army, was half US (VI Corps) and half British (X Corps), one Corps each under General Mark Clark, until the US Corps was reinforced by the US II Corp, when VI Corps were withdrawn and replaced by the French Expeditionary Corps.
The French Expeditionary Corps numbered 122,000 French troops - 60% Magrebhis, 40% French (those mostly from North Africa). They'd already fought in and liberated Corsica. After Rome fell, the Corps was absorbed into the French First Army, (4 French Divisions with 4 North African Divisions and 1 North African Brigade-equivalent with still over 50% Maghrebi soldiers in total).
The 3rd Algerian Division (the most successful of any French division in WW2, but with very heavy losses) was just one of the divisions (with 2 Moroccan divisions fighting with them in Italy), fighting on in the invasion of Southern France, liberating Toulon, Marseilles and Cannes, then into the Vosges Mountains fighting through the Colmar Pocket. They cleared Strasbourg and had more fighting in Alsace, before they fought as part of the French II Corps, spearheading that Corps' travel through the Siegfried line, taking the XVIII SS Corps in the Black Forest and then Karlsruhe and Stuttgart.
The 3rd Algerian Division's traditions and battle honours are (rightly) proudly carried on by the 3rd Mechanised Brigade today, garrisoned at Clermont-Ferrand, although some of its regiments are at different locations.
By 1945, the Indian Army was 2.5 million men strong. Of this, over 600,000 were Muslim. Naturally most were occupied in India and Burma, but Indian Army Divisions took part in campaigns in France, East Africa, North Africa, Syria, Tunisia, Malaya, Burma, Greece, Sicily and Italy.
It's recorded that there were 15,000 Muslims in the US Army in WW2. There were of course Muslims in the British Forces (as distinct from the Indian Army), including those perhaps less-known outside the UK such as Noor Enayat Khan, to say nothing of massive numbers in the Merchant Marine.
I wouldn't want to make a guess to what extent which forces were responsible for what was clearly a joint effort that depended simultaneously on many groups. For example, it would have been very different without the Soviet Union - could you say they had a hand in the liberation of France? Since Britain did this, or the US did that...and so on, as infinitum. As Dejw says, it was a World War, after all.