Here’s a whole thread from 2014 on mostly the same topic as this thread that quickly morphs into your specific question.
Short answer would be the C-152 or equivalent is far faster. One switch on, turn the key, waggle a lever or two while the stater is cranking and she fires. Once the engine is running push the throttle to full bore and be airborne in another 30ish seconds.
Pretty much any non-fighter jet will be about the same total overhead, plus the same time interval per engine. Which adds up to 3-5 minutes total. So any 2-engine jet will be about the same as any other, and any will be faster than any 4-engine jet by the time it takes to start the 2 extra engines. SO of your examples, a generic pre-MAX 737 and the Gulfstream would be effectively a dead heat.
Modulo the extra start delay built into the MAX and some Airbus NEOs that I mentioned in my previous post. So the MAX specifically will lose to the G550 for that reason.
Another advantage of the Cessna is it uses a much shorter takeoff roll and so could use darn near any taxiway, even a stubby little “side street”, on an airport. The jets can certainly use a taxiway as a runway too, but it would need to be one of the longer straight runs, typically one that parallels a runway. Airports don’t generally have long straight ones running in other directions. So the odds favor the Cessna being able to more quickly get aligned to a long enough stretch of pavement.
You said once wheels up you’re safe. If we alter the game a little, the answer changes …
If the goal is to get out of walkie-talkie range of the bad guys because they have the remote detonator for the gizmo (or out of cannon range so you’re not shot down), then you need to factor in the flight speed.
The Cessna might be airborne 4 minutes before the jet. And in those 4 minutes will cover about 5 miles. Once the jet’s airborne it’ll cover those same 5 miles in barely another minute and still be accelerating. In a drag race to 10 miles away from the airport the jet will win every time.