Obviously, situations differ. If nobody else had looked after the kid, it would have been the priest’s job. But in general, if an emergency occurs in any public place, whoever is speaking or doing something is supposed to keep going, in order to prevent public panic and rumors. “The show must go on” is not a selfish thing to say; it’s responsible.
(Oh, and reading the Word of God at Mass doesn’t cease to be important if nobody is listening. If that were the case, every lector would have to be a telepath, and stop every five seconds as somebody’s attention wandered.)
But in general, if someone gets sick, it’s the ushers’ job to deal with it. They are the ones who are mobile, they are the ones who can direct emergency personnel and do crowd control. They can also pass messages to Father or the deacon or the servers.
Re: the situation with Mass continuing during treatment of heart attack victim – That’s totally normal. I can’t tell you how many times I’ve seen stretchers in church. But normally, if it gets to be close to Communion time, the ushers tell Father and they have people pause and pray longer than normal before Communion starts. That also allows Father to receive Communion himself, do some anointing of the sick and bring viaticum, and so forth.
(Of course, if you have more than one priest in the parish, Father up on the altar doesn’t have to do that, because the other priest has already been summoned from the rectory to take care of things.)
If it’s a weekday Mass and Father doesn’t have any servers or deacon to delegate, usually some of the men in the weekday Mass crowd take care of this sort of thing. (And very often, they are men who did their time as servers or ushers, and thus know the emergency procedures that the parish follows.)
Every parish should have emergency procedures and people who know what to do. It shouldn’t all rest on the priest’s shoulders, although obviously he should make sure the parish is set up for emergencies.
