Please correct me if I'm wrong but AFAIK there's a DFT and Network Rail and RDG agreement which says trains are accepted to have left one minute early on TOPs.
TRUST do not record seconds in actual performance (so 10:14:59 is recorded as 10:14, as is 10:14:01), although berth offsets are correct to seconds in calculating the arrival/departure times.
Once the dispatch process is complete, the service may leave, so yes, as far as TRUST records are concerned, departing one minute early is absolutely fine.
HOWEVER this does not mean it is acceptable to dispatch a service early under normal circumstances. If the advertised dispatch policy is T-30 then that is what should happen. The doors should not close at xx:xx:20.
WTT shows a booked departure time of 19:47:30, so while it shouldn't be presumed passengers know that, 30 seconds before departure should have been 19:47:00.
Dispatch always goes by the advertised GBTT times, in whole minutes. The advertised GBTT is what reflects customer expectation. A train advertised for 19:47 gets dispatched at 19:46:30 where T-30 applies, provided the road is clear.
I was halfway across the footbridge when I saw the train pull in, but didn't leg it as I thought I had plenty of time to get there comfortably by 19:36:30, perhaps just a guard that assumed it departed at :45 rather than :47 - looking at the timetable throughout the day it is :45 but then goes to 18:43, 19:47, 20:45, 21:46 and 22:48!
Absolutely. Not really excusable but force of habit is a surprisingly easy thing to fall into. We all make mistakes sometimes unfortunately.
When you contact Northern you could invite them to use CCTV to confirm your version of events - they might not bother doing so, but it's an invitation and gives weight to your allegation.
In my professional opinion, this has absolutely no impact.