Below is a typical model rocket launcher schematic.
Worth noting s how when continuity test is done we use a 2 K Ohm resistor to limit the current through the igniter. The idea here is to keep the current through the igniter below limits where the igniter gets hot. Once the user knows that continuity is OK pressing the Launch button bypasses the current limiting and the igniter gets full current and ignites the engine.
Working from the schematic we want to automate the process manually done using the 2 switches and the Launch button replacing the button with a MOSFET. One problem is your uC will only provide 3.3 volts to your MOSFET Gate. Rather than a MOSFET you may want to consider a transistor like the old 2N3055.
You can likely look at voltage drops across the 3 LEDs. PWR, ARM and READY. When all 3 conditions are met you start your 10 second countdown.
I would start by using a Google of Model Rocket Launcher Schematics as there are dozens of schematics out there. You can use a circuit as simple as the above or get as fancy as you wish. A Google of Model Rocket Launcher Arduino will yield circuits as used here in this example.
Ron