How does ATC handle changing QNH/altimeter values? Obviously for aircraft arriving to or departing from an airport/airspace (as relevant), they listen to the ATIS and/or are given the QNH.

But what happens if that pressure changes? Even a fairly small change can make a big difference when we're dealing with 1000ft of vertical separation (which can be much less by the time we account for human margins for error).

Is the new QNH announced to the entire frequency, or similarly the change of ATIS, which each aircraft would then be expected to check? Or would it just be updated when assigning a new altitude to any aircraft already in the airspace?
In the US, ATC simply broadcasts a notification on whatever frequency is needed. For the terminal environment, section 2-9-2 of the ATC orders says:


  Broadcast on all appropriate frequencies to advise aircraft of a
  change in the ATIS code/message.


The phraseology is usually something like this:


  Attention all aircraft, Bowman information Kilo is now current.


For en route operations (see section 2-7-2) the ARTCC gives altimeter setting updates directly:


  Attention all aircraft, the Nashville altimeter now 2992