Back in the day, just how often would a typical railroad worker need to have his watch serviced to keep it running within the prescribed guidelines?

Once every 18 months up here in Canada on the Canadian Pacific Railway.

Larry

Per Santa Fe rules of 1938, watches were inspected for timekeeping at two-week intervals, and serviced every fifteen months.
Regards,
Bob

How was the timekeeping checked? I assume the current time was compared with the previous check to see what the error was over two weeks. I've always wondered what was to keep the owner from cheating by re-setting the watch between checks.

The watch inspectors of the respective railways checked the watches and kept a record and the railway employees had a card that showed the inspection record.

There would be no benefit in cheating.

The result of cheating could be a train collision with at least financial loss and perhaps the loss of life.