The simple answer is, usually the altimeter setting does not change fast enough for it to be an issue.
Near ground level, one-hundredth of an inch of mercury (00.01 inHg) corresponds to roughly ten feet of pressure altitude. Thus one inch of mercury corresponds to one thousand feet of altitude. (This rough rule of thumb can be seen by looking at a lowest-usable-flight-level table: Below 2992 FL180 is unusable, below 2892 FL190 unusable, below 2792 FL200 unusable.) This is close to negligible for separation purposes, given that minimum IFR-IFR vertical separation is 1000 feet and ATC scopes only display altitude to the nearest 100 feet. Of course arrival aircraft are more concerned with having the latest altimeter setting, especially if the ceilings are close to instrument approach minimums!
If the atmospheric pressure is changing quickly, the weather station will note that with the remark PRESRR (rising rapidly) or PRESFR (falling rapidly). This provides a heads-up to pilots and ATC to keep an eye on the setting. What is "rapidly"? 00.06 inHg (sixty feet) per hour! Not a gigantic difference. It's not super rare to see this at my facility but it's not an everyday occurrence by any means.
When aircraft are enroute below the flight levels a controller should advise them of the nearby altimeter setting "at least one time when operating within [their] area of jurisdiction" (7110.65 2–7–2 c1). Essentially every time an aircraft is switched to a new facility they will get the local altimeter setting, and by listening when new arrivals come onto the frequency they can update as needed. If an aircraft is showing drastically off (300+ feet) from their assigned altitude, ATC will re-issue them the altimeter and ask them to confirm their indicated altitude.