Odd. My understanding is that the computer cuts power to the pumps after a couple seconds if it doesn't get indication the engine is running. So I don't know why it would kick in after 30-60 seconds.

Fuel pump relays do fail, with cracked/failed solder joints. I can see a scenario where it just almost makes contact and finally does - or where it warms up due to coil flow inside the relay, heat expansion forces the cracked joint to meet and close.

Fuel pump relay should be on passenger side kick panel. As shown in fuzzy pic below, there are six parallel contact blades on a white relay housing. Angled nibs make sure the connector goes on the right way.

You can reflow the solder joints inside, or replace the relay.

To test: Ign. can be off for this. You can bypass the relay by jumping from fuse #4 to #6. Left or right side is OK but I think left is easier to get at. With jumper in place the pumps should run. If they run immediately when jumping but delay when turning the key then I'd suspect the relay.

Main pump under rear seat, drivers side - will often fail intermittently before failing completely. Usually doesn't kick in after a delay but will often kick in if you tap moderately on the pump "cage" with a small hammer while key is held in "start" position. Don't hit it too much or you can kill a good pump. Watch where you hit, there are wires and fuel lines there of course.
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Sven: '89 245 NA, 951 ECU, expanded air dam, forward belly pan reaches oem belly pan, open-front airbox, E-fan, 205/65-15 at 50 psi, IPD sways, no a/c-p/s belt, E-Codes, amber front corner reflectors, quad horns, tach, small clock. Wifemobile '89 245