Discounts will be offered from 1 to 15 March
Discounts will be offered from the beginning of March until / to March 15

I think any of these would be OK UK.
'through' definitely US

"Through" is definitely only US English. Since the terrorist attacks in the US, UK English has adopted the US order of "September 11 attacks" etc, but otherwise it's still much more common to write the number first (and, although you don't ask about this, the UK order would be DD/MM/YYYY i.e. 04.03.2003 rather than the US MM/DD/YYYY order)

HTH

I'm beginning to feel a little old-fashioned as it seems that I'm in the minority by still using formats such as: "from 1st March to 5th March" though I'm equally happy with "from March 1st to March 5th"

If I write a date in full I use "1st March 2003" or "March 1st, 2003" probably with equal frequency (though I baffled as to why I need the comma in the latter case).

Overall I think that it's safe to say that in the UK the writing of dates is undergoing change away from this answer, creating a state of flux in which almost anything is possible as long as our preference for "to" over "through" is respected.

is a different way of expressing it, if, as I understand it, both 1st March and 15th March are inclusive.

HTH