The London and Croydon Atmospheric Railway opened in 1846, although the experiment failed within a year. Carriages were propelled along the line between Croydon and Forest Hill by a metal vacuum pipe which ran between the rails. A piston ran from the train to the pipe, and was pulled along by the vacuum created by pumps at either end. Since the engines were at the extremities of the line, travel between those points was quiet and clean.
However, to maintain the vacuum as the carriages moved along the pipe, the piston had to run between leather flaps which formed a seal in front and behind. That relied upon a great deal of flexibility in the leather, and in practice there were frequent problems both with the pumping stations and in maintaining the seal. The appeal of the greased leather to rats didn't help! As a result, the experiment was abandoned. A section of the pipe can still be seen in the Museum of Croydon.
The line was soon absorbed into the regular railway, but parts of the West Croydon engine house made their way to the Surrey Street waterworks pumping station. It's not certain how much of the railway building was brought here: the listing text suggests that it may be no more than a few quoins (corner stones), while other sources suggest that large parts of the fabric were transported here and reused, albeit to a new design. The dark brick building of 1851 was extended in 1862 with the addition of a polychromatic brick building (to the right in this picture) and further enlarged thereafter. Today, it is empty and awaiting redevelopment.
The Croydon experiment may not have succeeded but the idea had not died by the time the Metropolitan Line was being planned. In fact, another kind of atmospheric railway would be constructed at Crystal Palace in 1864. This time there was no awkward little pipe with its difficult-to-maintain seal. Instead, the whole carriage fitted inside the vacuum tube; a collar of bristles at its end kept the seal. It seems to have been successful, encouraging construction of a longer, more serious project: the Waterloo and Whitehall Pneumatic Railway. The Whitehall project would never be completed - a recession and banking crisis meant that funding ran out. It's a fascinating story, told in much more detail in an ebook by Ian of IanVisits.
A third scheme using stationary engines was a cable railway which had operated since 1840, between Minories and Blackwall. Its carriages were attached to a rope cable, and pulled along by an engine at each end. One carriage served each intermediate station, so all journeys had to be made via one of the termini. That doesn't seem to have put passengers off, and the change to steam locomotives in 1848 was made largely because of the cost of replacing worn ropes. The railway only finally closed in the mid-twentieth century. Today, parts of its line are used by the Docklands Light Railway.