There are two different things here, and no, you don't particularly need to worry or get online at an inconvenient time, though of course (a) it's a good idea to check-in early online for all sorts of reasons and (b) regardless, we are all at risk of seat shifting on any aircraft, any airline, stuff happens.

What I believe JAXBA is referring to here is that if you didn't have a seat reservation - e.g. through lack of status - you may well have an underlying, usually invisible, nominal seat allocation. Sometimes, on the BA App or other software tools, you may get an inadvertent glimpse of the seat number. And this is fairly transient in nature, particularly when Theoretical Seating has crunched its numbers. You are in a different group, as if you had a paid seat, with a firm allocation which in normal circumstances won't change.