It also depends on how good/bad/new/old the gauges and/or sensors are, it can vary wildly from aircraft to aircraft.
In theory if we don't see oil pressure rising within 30 seconds of start we should shut down, but in practice (on many of the battered old ruins of training aircraft nowadays), it could take until you're doing your power checks to see a decent reading. It can be helpful to ask the instructors familiar with that particular aircraft the question you've posed here.
Immediately after start, it's the starter warning light that's the BIG biggy.