I’m confused whether the latter can be used or not. Would you tell me?
‘thirty one July.’ might be correct in foreign varieties of English but, it's not in the English kind of English. In England and the rest of the UK, it would be exceptional to find even one person who said dates like that. I haven't heard that at all. It could be that saying it that way is something peculiar to that, particular person (Jim Dale).
In England and the rest of the UK, dates are normally spoken in the English way which is how he first reader (Stephen Fry) said it, ‘the thirty first of July,’. They are in the order of day, month and year.
Unless you use a particular, foreign variety of English where ‘thirty one July.’ would be correct; it would not be used.