Today is Saturday the second of September

Yes, that is correct. Alternatively, one might say: "Today is Saturday, September second."

British speakers overwhelmingly use 'the' before the ordinal number (the first, the second, etc) when speaking dates aloud in that fashion. They might very often put the ordinal before, and 'of' after, the month name: Saturday the second of September. When writing dates using figures, there is an increasing tendency to omit the ordinal suffixes -st, -nd, -rd, -th, so instead of "2nd September, we often write "2 September", although we would never say "two September".

I meant to write "They might very often write 'the' before, and 'of' after, the ordinal number: Saturday the second of September."

That is how you hear/use it in Canada.