Here are a few tidbits, largely from "How Steam Locomotives Really Work" (Semmens and Goldfinch, 2000, p 163-) and "Premier Line" (O.S.Nock, 1952, p80-) on the LNWR's story with compunding:
The main reason that compound engines were not pursued in the long term is that good quality coal was quite plentiful in Britain (as opposed to France where they were developed through to the 1930s), and thus there wasn't as critical need to make compounding work. The efficiencies achieved (contemporaries in Britain suggested around 10%) could also be found through superheating, or improvements in steam flow and valve gears. Francis Webb started building three-cylinder compounds for the LNWR at a time when Richard Moon's chairmanship prescribed relatively low speed express services for maximum fuel efficiency.
His design used uncoupled driving wheels, with a low pressure cylinder driving one axle, and the other driven by the two high pressure cylinders. Each set of cylinders had its own reversing gear. When setting off from a stand all the work would be done by the high pressure cylinders, which were significantly smaller than earlier locomotives (those on 'Experiment' had less than half the area of the earlier 'Precendent' class). As the driving wheels were uncoupled as well, wheelslip was also a danger. Once they got going the low pressure cylinder would start working, but there was still the possibility of one (uncoupled) axle or the other slipping, which would lead to the receiver (taking exhaust steam from the high pressure cylinders to feed the low) either being flooded or drained.
Unfortunately for the 'Experiment's, the timetables of the LNWR were sped up just as they were introduced, and while respectable at moderate speeds they required "a tremendous lot of hard flogging to get much over 50 or 55 miles an hour out of them", suggesting a restricted steam flow. Their successors, the larger 'Dreadnoughts' (still three-cylinder compunds with unlinked driving wheels) had less of an issue with wheel slippage due to the greater adhesive weight and were capable of higher speeds (up to 70mph) and greater loads, but were still heavy consumers of coal due to poor steam flow.
The later 'Teutonic' class differed from the Dreadnoughts in having a single reversing gear, with the low pressure cylinder driven by a loose slip eccentric mounted on the crank axle. This meant that the low pressure cylinder was always in full gear (for maximum steam flow throughout), but it also meant that the low pressure cylinder's drive wasn't reversed until the locomotive had been running in the new direction for a few feet. A valve could be opened to allow the high pressure exhaust to bypass the low cylinder when reversing, such as after a locomotive set back onto its rake of carriages at Euston. If the valve happened to stick then one set of wheels would be turning backwards as it pulled off... but if the other set of driving wheels should happen to slip then there was the comic scene of one set of driving wheels spinning forward, the other spinning backward, and the train going nowhere!
Webb moved on to four-cylinder compounds (two smaller low-pressure cylinders), and this time had them all driving the same axle, with coupling rods linking the other wheels. Wheel slipping was avoided, but with the valve gears linked up again the steam flow on these locomotives weren't as free running as the Teutonics with their full gear low-pressure cylinders. Webb was now a giant in the world of engineering, but never managed to iron out all the kinks in his engines, and in his later years simply ignored his critics. The fact that his compound locomotives were poor performers on the most prominent railway service, the express passenger train, led the general manager Sir Frederick Harrison to lay down a general instruction that every train with a load greater than "17" to be double-headed, whatever the class of locomotive used. This preserved the LNWR's reputation of punctuality, but was a harsh condemnation of Webb's locomotives.
After Webb was ousted in 1903, his replacement, George Whale, initially sought to address the failings of the compound locomotives. He fitted separate valve gear for the high and low pressure cylinders of the four-cylinder compounds, which allowed them to be operated in the same way as the Teutonics, although they weren't quite as free running as their three-cylinder colleagues. The introduction of a simple (non-compound) 4-4-0 'Precursor' in 1904 surprised many in breaking away from compound technology, but the simplified design was able to haul heavy loads at high speed with ease won everyone over. New Precursors were being built in the workshop as quickly as the old three-cylinder compounds could be cut up, and the LNWR stuck to single expansion engines from then on.
In summary, the LNWR never managed to perfect the more complicated compound engine, and the alternative of a simple engine that 'just worked' won the day instead. I understand that the Midland had slightly better success with compounds, using two low pressure cylinders and one high pressure cylinder, and at least one batch of these was built by the LMS, but I don't know any details about them. It's possible that the reputation of Webb's compounds may have discouraged some others from experimenting with them at all.
