Creating an electric alarm
This is mostly a mechanical problem if you intend to start the alarm timer automatically. As we don’t have any information on that, I’ll assume that there is either a low voltage contact closure or mains is applied when the timer is supposed to start.
So now the problem is reduced to making an alarm indication 10 minutes after the start.
An easy solution would be to use an Arduino board. You could either program the board using the Arduino IDE or
or use something like the gcc tool chain  to write a bare metal program for the AVR chip.
All the program has to do is delay for 10 minutes and the generate whatever indicator you want, such as a port pin changing state to drive a relay through a transistor or a 2kHz signal to drive a beeper.
The program could be as simple as using library delay and PWM functions, or you could delve deep into the datasheet and use timer peripherals and interrupts. Adding a countdown display might be nice, though it starts to look a bit suspicious.