Hello Jonny,
The non-heating could be one of a few things(..!), I don't know how confident you are taking the Classic apart but here's what I'm thinking,
As GreenBean said, the thermal fuse could have seen triggered (this is on top of the boiler, usually in the middle under a metal clip, housed inside a ~3 cm long plastic tube). I don't know the test for this but I'm sure a continuity test would suffice. If this is the issue you will need to buy another and fit it. I don't think the Classic has had a re-setable thermal fuse before but if it does, it will look like a little button set in something roughly the size of a 5 p and will most likely be made from plastic and will be situated somewhere on the top of the boiler. If you do have one, press the button in. If this was the issue you will hear and feel it click. If you press it again you will not feel anything. If this was the issue, phew! It's probably sorted.
OR
The heating element could have broken, unfortunately the elements are integral to the boiler and will require a new boiler if this is the case (as far as I know). The up side to this is that the new boilers are very shiny
Again, a continuity test of the heating elements will probably to the trick. If you look at the boiler you will see that there are 4 pins on the top connected to wires, they are the connections to the 2 heater elements, they are long 'U' shapes on either side of the boiler, made from metal and you will see which 2 pins to test. As I understand it, you should be able to test if you have lost a heater element by flicking the machine in to steam mode. This changes the 2 elements from series to parallel. One should get hot, one (the one breaking the circuit) will not heat up.
OR
The lower temperature thermostat has broken. I'm not sure what happens to these when they break but I'm hoping that they don't allow the machine to heat up! I think if you flick it over to steam mode it should use the other thermostat (to make steam) as feedback and bypass this possible broken thermostat.
If you need any of these describing in any more detail just ask - I can do it to my machine (it's a Coffee but these steps will be the same) first and show you with photos if needed.
I hope you get it sorted!