If you are asking about British custom and practice, it's usual to alternate words and numbers in a date: Thursday 23 June 2004, not Thursday June 23 2004. It's also common to omit commas in a date, particularly when it's part of another sentence which needs its own commas. You don't need full-stops in pm, but there is no harm in using them either.
I would write
On Thursday 23 June 2004 at 3:15pm, Mike Lollis resigned.
Actually, I wouldn't write that; I'd put the second clause first. But anyway...
Using a thin space in the time (before pm) would be an improvement — particularly when using non-lining figures — but that's not easy to achieve in normal settings.
You could put a comma after 2004 which would turn "at 3:15pm" into a parenthetical insertion:
On Thursday 23 June 2004, at 3:15pm, Mike Lollis resigned.
If you want a comma after Thursday it would be better to have another after 2004:
On Thursday, 23 June 2004, at 3:15pm, Mike Lollis resigned.
I would have no problem with a liberal scattering of punctuation as your edit suggests:
On Thursday, 23 June 2004, at 3:15 p.m., Mike Lollis resigned.
You can use as little punctuation as is grammatically required (just a single comma) or as much as might be justified (three commas and the abbreviation full-stops). What is necessary for British practice is to ensure that the day and date come before the month.