I don't think there would have been 'radical' differences if the US and Brits/Canadians/Poles were on the other side.

The great swan from Normandy saw the 21st AG race like a bat out of hell to the Low countries.

As for the number of vehicles that is of less importance. During the 'swan' the recce and Armd brigades led the way. The British drew upon lessons from the MEF/8th Army lessons in 1940-42. Immobilising some units to give 'fresh' units right of way.

As an example my pet div (51st Highland) stopped at St. Valery where the original Highland Div had been shot to bits by Rommel in 1940, whilst other divisions took the van northwards.

A serious problem in the campaign was to get enough rescources to the fighting men. Even with the PLUTO in operation convoys needed to travel across France to supply the front.

Monty wanted the Narrow front to ease the supply situation, but we don't need to go down that lane do we?

To sum up. No I don't think that the reversal of forces would have done much difference.

In 1944 the British army was finally coming to terms with fighting efficently. However most revisionists give a similar description of the Brits in '44 as they would in '41. This is unfair as the Army did no less than 3 doctrinal revisions in that space of time.

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'We march. The enemy is retreating in transport. We follow on foot.' Lt.Neil McCallum 5/7 Gordons 19th November 1942