Today is women's qualifications.
There are 5 subdivisions, with about 20 gymnasts in each one, who go around the four pieces of apparatus.
The first two subdivisions were this morning, and were shown on BBC - Team GB was in the first one, so that was shown; and the US were in the second one, with Simone Biles, so that was also shown.
The third one is happening at the moment, but unless you have a paid-for streaming service, you can't really watch it - you can keep updated on the scores on the olympic website, though.
The fourth and fifth ones are later this afternoon and this evening. At the end of all 5, it will be clear who has qualified for the all around final (the top 24 gymnasts who did all the apparatus), the event finals (the top 8 in each piece), and the team finals (the top 8 teams) - however only 2 people can go in the individual finals/event finals from each country, so that pushes some of the lower ranked finalists up into the final.
So it'll be later tonight when the final lists are known. But some things are obvious already - e.g., the US has already qualified as a team, as there are not enough teams left that could overtake them and leave them without a spot. It's also pretty obvious who some of the top finishers in all-around and event finals are, but some of the places are still up for contention.
I doubt much more of the qualifications will be shown on BBC, as the British interest in finishes for the day and so is Biles, but they might show some of the final subdivision with people like Rebeca Andrade in (a main competitor to Biles), or a few of the others.
Britain will possibly have a couple of gymnasts in the all-around, and might qualify as a team and might have a gymnast in one of the event finals (bars), although none of those are certain. The team finals are on Tuesday, the all-around on Thursday, and the event finals over next weekend and Monday.
That's just women's artistic. I haven't really followed men's, or the other formats.