Hi mate. What model year / car / engine size / 8v / 16v do you have?
Its hard to say without that info!!!
There are numerous faults that can cause this behaviour espec on 95 to 98 gti mk3's, (2.0 8v)
relay 30 (ecu power), in fusebox pos 3
relay 167 (fuel pump power), in fusebox pos 12
Hall sensor, near dizzy
crank position/speed sensor, near dizzy
oil pressure sensor, under oil filter behind front engine mount
Throttle body full of crud, TPS (throttle position sensor) faulty,
MAF sensor, on airbox exit
ISV (idle stabilisation valve) if fitted, near oil filter i think, 8v motors dont have this one.
Search the forum for "cutting out" or "cuts out" and see the many 1000's of posts on the same fault....can be just chafed wires to the sensor or failure of the sensor. The ecu may go into "safe mode" causing auto box weirdness due to a sensor error.
1. does the oil light flash?
2. does it cut out while driving at speed, or just when idling?
10 minute delay may be overheating relays.