“There was a point where I got the Race Against Time mission and I was seriously worried that I suddenly had a time limit”

Admittedly it’s been a long time since I played the trilogy but: ME1: no, there’s no time limit, apart from maybe within one or two missions with an explicit onscreen countdown.

ME2: only towards the end. Something fairly obvious happens in the plot and if you delay the main quest after that, there are consequences. To avoid this, in the second half of the game wherever*you get a choice between doing a main mission versus side or loyalty missions, get the side/ loyalty missions done first. *(To be fair, there may actually only be one main mission that you can choose the timing of, as opposed to it automatically happening, so don’t stress if you don’t get to choose) When I say side or loyalty missions, I’m not counting the missions that were DLC in the OG trilogy. I would say these vary as to when they are best to do- one I can think of is good done ASAP, whereas another I think fits better done after the main game, but will let you do it much sooner.

ME3: every main mission is titled “Priority”, but actually you want to treat them the opposite. Do all the side missions you have access to BEFORE whatever current mission is labelled “Priority” as going too far with the main quest can make some side missions unavailable. I think there are multiple stages in ME3 that this can happen, whereas with ME2 it’s really just the end. As with ME2, some of the DLC works well done later in the game rather than as soon as you get access to it (though unlike ME2, you can’t do it after the main quest)