When you say the button do you mean the main oven control knob?
Can you turn the oven control knob?
Yes the main oven control knob. The knob still rotates but will not push in to activate the electric starter
Ok im with you.
If the dial isn't pushing in then im afraid its more then likely a faulty oven thermostat as with it being an auto ignition its part of the thermostat and when they sieze up like this its a case of replacing the thermostat.
What you can try as the very odd time its worked is remove the knob by pulling it off and spray a bit of WD40 onto the thermostat shaft and see if the frees it off.
But if still the same then im afraid its a case or replacing the thermostat in the oven to get this working again.
How much should you expect to pay to get this issue sorted by an engineer?
Ok the part itself is £46 from Hotpoint and roughly for labour you are looking at about the £50 mark maybe more if you are in London area.