How to replace a manual timer by a simple on/off switch on a home appliance
Get a slightly different timer

You are correct; a programmed electronic timer is a bad plan because it will need to power itself, and that will require additional wires (e.g. Neutral) that may not be present.  

What I recommend, instead, is a different mechanical timer with a stay-on zone.  This looks exactly like the timer you have, but for an additional zone past the maximum duration, which says "Hold" or "Remain on".  Wound to this position, it will not unwind.  

I generally don't use that type, my users are idiots, so I use the type that makes it impossible to stick the thing on continuously.  That is why they market both types.  

For instance, in the States, the standard brand is Intermatic and is available in both types in a variety of durations.  If you're unhappy with the choice of duration range, this is also a good time to solve it. 

Abuse the one you've got

One thing about those mechanical timers: you can't put "just a tiny amount" of time on them.  If you have a 1 hour timer, it's clearly labeled that you need to wind it past 12 minutes (and then you can wind it back to 3 minutes).  If you only wind it to 3 minutes,  haven't put enough preload force into it for the snap-off  mechanism to work, and it will stick there and never shut off.  

Feature :)