You can get a time delay relay, but it may not buy you anything. You'd probably get less shock to the system if the pump came on when the boiler lit. In fact, for max efficiency, the pump often runs when the boiler isn't lit.
For a boiler that old, it would be hard to tell what is going on,especially without any schematics.
On a modern boiler, there are numerous safety interlocks.
- minimum pressure to ensure you won't flash water to steam
- flame sensor to ensure you aren't running fuel but it is not lit
- max temp to shut it down if it gets too hot
This is not counting the mechanical things like a T&P valve.
Hard to say what yours has on it.
There may be a flame sensor on the gas valve that is shot...it tries to light and thinks the burner did not start, so it shuts off. Modern closed combustion boilers then go into a post purge to remove all fuel vapors, and then often will try again. A modern controller won't keep doing this forever, and gives up after a few tries. They often do a pre-burn purge, too, just to be safe.
I'm not a pro, but I'd look at the thermocouple that senses the flame is on...seems like it is trying to start, but doesn't think the burner actually lit, assuming yours has one.