Firstly, no, subsequent AIs will not mess up any previous AI which held - unless you are using hormonal interventions, in which case I don't know.
Secondly, these are Jerseys, I think? Two of my Jerseys are due 16th March and 22nd March, and neither are really bagging up yet. Just about to start, I think.
I think Jerseys start to bag up a lot sooner when they're heifers - as a calving heifer, Hillie had us on the watch over 4 weeks (we didn't have a confirmed date, just a 'not before'), and had had a bit of a bag for nearly 3 months before she calved!
Our suckler cows often show little bag - we've one at the moment, she's a British Blue cross, 1/4 dairy and 1/4 Angus. She has no bag to speak of, but she's clearly giving plenty of milk. Mostly the half-dairy ones do show a bag (sometimes too large for one calf so we have to get set-ons to use up the excess milk) but often the 1/4 and 1/8 dairies appear to have little bag - but they all rear fine calves.