Americans almost never speak that way. We would say "March the Second, Two Thousand and Three" or "March Second, Two Thousand and Three". We would never say "Two March Two Thousand and Three".
The people who use the "2 March 2003" style would say "Second March, 2003", not "Two March". It's natural to British speech (or, at least, none of my British friends seem to find it artificial when they use it).
you could say "the second of march, two thousand three". Sometimes I say it that way.
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