Note: I tried to replace the dc/dc converter with an LED and the LED turns on based on the delay (however the voltage across the capacitor remains 0.75-0.8V)
Of course, it is clamped by the base.
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If you really want to use a NPN part, then work back from the 50mA. Ballpark might be 1v change and 1mA base current, to saturate reasonably at 50mA.
That puts C as ~3mF or 3000uF, and R as ~ 2k2
Or, you might want to change to a MOSFET - that has no base current limitation, so you can use higher R and smaller C.
You need a lower threshold gate mosfet that will drive 50mA from 3V, so a FDV303N or similar part, that gives < 1 Ohm with VGS 2.7 or 2.5V