Manchester, UK Airport time to get through security

I can't comment on how long check-in/security would take, but depending on luggage, it's at least ten minutes from the station platform to the departures lounge in the terminal, and quite possibly 20 minutes.

Manchester Airport has a single, central rail station connected by long passageways with moving walkways to the terminals, so it's roughly comparable to changing terminal at most airports.

Standard guidelines state that for European flights you should aim to check-in two hours prior to departure. Whereas for domestic flights the Manchester airport suggests one hour. I think something in between is a good tentative plan, especially if you have luggage to check-in.

Regarding the transfer time from the train station to the British Airways gate, it's hard to be exact since security checks can take an arbitrary amount of time, depending on how busy they are. Nevertheless, let's try an educated guesstimate.

For the station to airport transfer, I ran a quick search using this nifty tool by inserting the information you provided. According to the information I got:

Train

The Station is a 10 minute walk from Terminal 1 and 15 minutes from Terminal 2 using our Skylink moving walkways. It takes about 5 minutes to walk from Terminal 1 to Terminal 3.

Considering this, plus the two-hour guideline, I would say that you should aim to get to MAN airport train station two-and-a-half hours before your flight, i.e. at 12:30 at the latest.

You could reduce this safety margin by checking-in online, which will should allow you to avoid the check-in queue and go directly to the luggage drop-off point. Note however that the queue-jumping advantage is airport-dependent. I.e. I don't know if at Manchester Airport BA provides different lanes for people who just have to drop their luggage off.

In case you don't have to check-in your luggage you could reduce the two-and-a-half hour margin even more. Of course these techniques are risky and could cost you your flight. Personally I am one of those people who prefer to be safe rather than sorry.

The reference point is check in 14:15 to allow the minimum 45 minutes interval between check-in and flight ETD:

Assuming you check-in on line, bag drop off, Security, transfer to gate etc is covered within that ¾hr. Once in the system just going with the flow should be sufficient.

I expect departure from T3 (worst case anyway, as it happens), so 15 minutes from train station:

hence need to be at train station by 14:00. Allow a margin (I suggest 30 minutes) for the sort of things that quite often don’t go wholly according to plan and that requires arriving by train by 13:30 at the latest.

It gives at least 1/2hr contingency over 1-1/2hrs, or 50% more than required were everything to go smoothly. And then add to that any extra safety margin from arriving a little before 13:30 because that is the last train scheduled to do so before then. The slack is as much for peace of mind en route as of necessity. Anything goes badly wrong and two or three times as much slack might not be enough anyway. The chances of every possible little thing going wrong to add a few minutes here and a few minutes there at every stage, and sum to a significant total, are negligible.

I’d certainly allow more margin if say the trip to Heathrow was to catch an onward international flight (specially if on a non-flex ticket!) But it seems the downside is nowhere near as bad as it might be (eg visa expiring that day). BA may have two later flights the same day to choose from (17:20 and 19:00) and even a single ticket is not cripplingly expensive (under £200). Plus there are trains, or even a taxi (about £230 - but about two hours longer than by air).

The trade-off is very roughly: waste an hour or so on every such journey to be reasonably sure even the worst plausible eventuality will not throw you off course – or save 12 hours over a dozen
such journeys in the expectation that one of those may well end up costing about £200 more than the others and take two hours longer. In other words, is an hour of your time worth £20 to you?

If yes: Get to the train station by 13:30
If no: Get there about 12:30