I assume you mean an N-channel enhancement mode MOSFET, since the other three major classes of MOSFET are even less suited to this circuit. (You made me guess by using a generic FET symbol here instead of a proper MOSFET symbol.)
Even with the "proper" type of MOSFET here, you're right, it won't work.
The core issue here is that a PUT lets you set a high trip voltage for this circuit, which is high enough to light the LED, as the PUT dumps the 2.2μF cap's charge through the LED briefly before the PUT turns back off.
MOSFETs simply don't behave that way.
For one thing, the MOSFET's threshold voltage — VGS(th) — will be well under the LED's forward voltage — Vf — so even if it does turn on, it will do so below the point where the LED can turn on.
I think you'd want the cap and resistor swapped, too, but that's a relatively minor point.
You can get to where you want to go from here by adding parts. You could even build a "programmable threshold MOSFET with hysteresis" — PTMWH — and have something that, externally, operates like a PUT.
Or you can skip all that and use a PUT. :)