The correct English structure should be month first than date if we use the th form eg. January 1st, July 7th etc
Same for the UK and Ireland. "# of [month]" should be accepted as it's correct English.
Putting the name of the month before the date is a bit of an Americanism if I'm not mistaken.
I don't think so. I have often heard English people refer to a date like "December 7th, 1941," though when writing numbers, they will use day/month/year. Americans, on the other hand, will refer to a date such as "the fourth of July" or "the fifth of May," but will put month/day/year when using numbers.