The earliest use of push button technology was in the 1930s, where keypads were fitted to some switchboards to increase the efficiency of operators. Digits were stored in a sender for outpulsing and signalling used light and heavy +/- voltages. The standalone Keysender No 5 was also available to work in place of a dial.
The circuitry to produce DTMF tones was invented by Bell Labs in 1959 and push button (PB) phones were first launched in the USA in 1963. These phones were given the name Touchtone. The GPO, as always, not willing to just accept that the Yanks had got it right, produced a several prototype PB phones with a number of different layouts. The prototype in the second image above uses the most unusual 1441 layout, though the US 3331 layout was also trialled, as below, another prototype.
In the end, however, when the first PB phones were launched, they used the 5+5 layout and this persisted for some time, including the first true DTMF phone, the 724, in 1966. For many years, the PO tried to insist on the use of the term ‘press button’, but this was not entirely successful!