Each year in Canada we celebrate our national day on July 1 and then a few days later, our neighbours to the south celebrate their own day. There are different ways to talk about dates, but in the American case, the holiday is almost always "the fourth of July". It could, however, be known as "July the fourth" or just "July fourth".
In the BYU Corpus of American English, the [MONTH ORD] to [the ORD of MONTH] ratio is about 7.3 to 1 for January (the [MONTH the ORD] is quite rare). We do, however, find [the ORD of MONTH] pattern for most days, with Jan 1 and Jan 15 being the most common of these.
In July, the ratio is closer to 1.2 to 1, still in favour of the [MONTH ORD] wording. Most of the difference is accounted for by the fourth of July with the first of July also scoring highly.
Now looking at the British National Corpus, the July ratio, 1.4 to 1, still favors [MONTH ORD], but here July 4 is only slightly ahead of July 12 & July 1. In January, the same ratio is 3.4 to 1, much less lopsided than the American ratio.
So, I think we can say that both formula are used on both sides of the Atlantic for all dates, that [MONTH ORD] is more common in both cases, that special dates are more likely to be highlighted using the less common wording, and that there is a stronger preference in the US for the [MONTH ORD] order.