Pulse dialling still works on many exchanges.
You could try something to see if it works. Use a short, but valid, free number like 155 (international operator). Don't use 111 or 112 as they are emergency services.
When a pulse telephone dials, it actually opens and closes a circuit at, roughly, 0.1 second intervals. Dial one, and it will open and close the circuit once. Dial two and it will do it twice. That's the clicking you can hear.
The same circuit is also opened and closed by the two black sticky-out bits that you put the receiver down on. So, to dial one, you can quickly press those down and release them instantly. Like tapping them.
So, to dial 155, you tap it down once, wait a second, tap it five times quickly (like 0.1 sec apart but it isn't too critical), wait another second and then tap it another five times. If it then starts ringing the International Operator, put the phone down quickly and feel confident that you can still use pulse dialling (and that it is a phone problem, rather than an exchange problem)
(If anyone else fancies trying that, you can all do it on the switch you put your receiver down onto. Remember to tap ten times for '0'. It can be a fun party trick)