I would like to replace a regular switch with a timer switch. The timer switch I have purchased requires a neutral wire. Looking at the existing switch (first photo), I have a black and a white wire connected to it, but I believe they technically should both be black, and I'll need to connect to the properly-labelled white neutral wires in the neighbouring box (second photo). I've attached a photo of the current configuration and a link to the manual for the timer switch. Any help would be appreciated.
Get a different timer
You should be able to get a different timer switch that does not need a neutral to operate -- any Leviton LTT series model will do the job in your case, for instance.