First thing...the proper terminology is "time play," not "timing play." It may seem like a nit, but I prefer that we all know what the proper terms are and how to use them.
Second thing...the confusion often arises due to the manner in which a runner is put out on appeal and on a force. An appeal out can occur by tagging the runner or the base where the infraction occurred. A force out can occur by tagging the runner or the base to which he is forced. It is the tagging of the base that gets people.
Difference between force play and time play: When it comes to scoring a run on the third out, it is always one or the other. A force play is any play on a runner before safely reaching the base to which he is forced. A time play is any non-force play that occurs and the status of a scored run is in question due to when it scores in relation to the out recorded.
We can move to definitions, which are the same in all codes. A force is when a runner loses the right to occupy a base because the batter becomes a runner. The batter has to go somewhere (1B,) so R1 will always be forced in these situations. A force is removed under two circumstances: when a succeeding runner is put out, or the runner reaches his advance base.
With that in mind, we can think of an appeal play. Appeals can either be time plays or force plays, depending on the infraction. The best way to think of it is to determine if it is a force play, and if it isn't, then you know it's a time play. Was the runner forced to the base where the infraction occurred? Were all succeeding runners still in play at the time of the infraction? If so, you've got a force play appeal. If not, it's a time play.
So, let's apply this to the OP. Was R1 forced to the base where the infraction occurred (1B)? No, R1 can only be forced to 2B. (We could stop there, because we know this is now a time play, but for illustrative purposes, I'll continue.) Were all succeeding runners in play at the time of infraction? No. BR was out on the catch (you may think the infraction occurred as soon as R1 left, but it actually didn't until the conditions were met, which is the legal catch of the batter's batted ball.)
So, in short, an appeal play is a time play unless it is a force play. A force of a runner at a base can only be at one particular base for each runner (his advance base,) which means that an appeal can only be a force if the infraction is for a runner missing his first advance base to which he is forced, and only if all succeeding runners are still alive when he misses it.