This is now a bit of a digression, but that’s not really how things panned out. There was no real problem in producing accurate timepieces for use by railroads without the use of electricity – battery or otherwise (as per the circular I showed in post #11). The problem (solved principally by the telegraph, which of course needed electricity) was synchronisation.
In the early days of railroad operation, companies had to cope with the fact that - because time was regulated according to the rising and setting of the sun – “local time” varied according to geography. For example, when the British GWR ran its first service from London to Bridgewater in June 1841, the published timetable carried the following warning:
LONDON TIME is kept at all the Stations on the Railway, which is about 4 minutes earlier than READING time; 5½ minutes before STEVENTON time; 7½ minutes before CIRENCESTER time; 8 minutes before CHIPPENHAM time; 11 minutes before BATH and BRISTOL time; and 14 minutes before BRIDGEWATER time.
The line was 156 miles long, so “local time” (solar time) for the daily lives of citizens varied by a few minutes all the way along the line. Anyone not taking this into account and relying on the local time displayed on their own watch risked missing the train.
Before the railways became established, it didn't matter that places at different longitudes kept their local time by the Sun since there was no means of transport faster than a horse. A few minutes discrepancy either way was of no consequence. The railway network allowing people to travel long distances in short times however needed a standardised time for scheduling purposes and to avoid collisions. It was the electric telegraph network that provided the solution. Not that the clocks used by railways needed to be electrically powered (although limited numbers of electric clocks did exist), but that the signals sent almost instantaneously via the telegraph enabled mechanical timepieces to be manually adjusted to maintain the synchronicity at all locations. Systems that enabled automatic synchronisation by telegraph signals (which did require the clocks to be electrically powered) eventually followed at a much later date.
In Britain “Greenwich Mean Time” (GMT) was adopted in 1847 as “Railway Time” for synchronisation purposes by the Railway Clearing House. By 1848 almost all railway companies adjusted their clocks and timetables accordingly. Then in 1852 a facility was set up at the Royal Observatory in Greenwich that transmitted accurate time signals throughout the country over the electric telegraph network. By 1855 nearly all public authorities, such as churches and town halls, set their clocks to 'Railway Time', displayed on station clocks by station masters who adjusted them according to the signals from Greenwich.
The adoption of standardisation elsewhere was a slow process that had implications beyond the railway network and was resisted in many places. Railways in the US and Canada didn’t coordinate their clocks until 1883, with some cities (Detroit for example) refusing to align public clocks to ‘Railway Time’. Then in 1884 at an International Telegraph Union conference in Washington DC, Britain convinced the other participants to adopt Greenwich as the prime meridian for measuring longitude and timekeeping. It took decades for full implementation. Uniform agreement wasn’t established in the US until 1918, when the ‘Standard Time Act’ entered the statute books.