It doesn't seem impossible, just difficult. After all, the industrial revolution was majority water powered for the first 40 years. They're already operating heavy machinery like pumps off of water wheels and using equipment like hydraulic accumulators so building up a hydraulic power network seems more an investment and economic issue than a purely technical one.
Perhaps in the 1830's or 40's a big steam engine explodes in London causing a coal fire that burns a large portion of the city. In the aftermath the public demands a safer alternative and Parliament decrees that factories in London must use hydraulic power instead of coal. The economic advantage of locking coal out results in the creation of several competing London hydraulic companies and a boost in turbine research. While expensive, the new system proves remarkably popular with the public since there's significantly less smoke, risk of fire, and fewer smokestacks around the city. Newspapers and yellow journalism jump on the bandwagon by declaring that London's health was improving and that coal smoke is clearly poisoning everyone, stoking a paranoia about coal. With London's success and the technological refinements coming out of it other places begin to adopt similar systems. "New Safety hydraulics instead of that nasty coal that will murder your family in a terrible fire". Several company factory towns in New England convert over. By 1850 or 60 it's common for the homes of the wealthy to be connected to this new water grid, providing both water but also power for the first in-home appliances. Urban heating and hot water are increasingly provided by a combination of this and the existing gas distribution system for lighting. City fires (which were a
huge problem in the 1800's) are increasingly rare. This in turn creates a whole new demand for canal and dam construction to provide more power and water.
At least that's one idea.
Coal is clearly has the advantage in areas where water and waterwheel suitable rivers are less common but the industrial base in this history is remaining much more confined to those initial areas. I think eventually coal-powered pumps for this new hydraulic system are almost inevitable but they're more like central power plants instead of everybody burning coal.
I did come across this crazy gem though while looking up turbines: It's in French but it's "Hydraulics applied. New system of locomotion on the railways." a method for fluid bearings and hydraulic propulsion for locomotives in 1853.
https://books.google.com/books?id=4CtWAAAAcAAJ