I would say it should warm up faster because the difference in temperature between the room and heater is higher.
Edit: I am talking about a convection heater.
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Because you're only changing the temperature at which the heater is supposed to stop working.
It is always working at the same power, regardless the temperature difference. But for higher temperature it will have to heat at this same power for longer time.
So in short: You don't change the difference in temperature between heater and the room.
The thermostat in a heater is usually an on-off device. It senses the room temperature and runs the heater at full power as long as the room is colder than the target temperature. If the room is hotter than the target temperature, the thermostat turns the heater off.
(In a narrow temperature interval around the target temperature the thermostat will usually stay in the state it had the last time the temperature was outside the target interval, such that it won't incessantly turn on and off based on fractions of degrees of difference).
Your description sounds like you're expecting the thermostat to be based on the temperature of the heating element inside the heater, but manufacturers do their best to avoid that and instead let it sense the actual air temperature in the room, since that it what you as the user actually have a preferrence for.
To add to the otherwise good answers...
The OP says in an edit that they are only talking about a convection heater. If the heater only has one heating element, then the answers about when the thermostat turns off are broadly true.
If the heater has multiple heating elements though, you can turn on more of them. Old-fashioned electric fires in the UK typically had up to 3 heating elements. The more heat the fire puts out, the faster you'll get up to temperature.
The same principle applies to central heating systems too. Most boilers allow you to set the temperature of the water circulating round the radiators. If you increase the temperature here, all your radiators will now be hotter and rooms will heat up more quickly. Do be warned though that this can be a real safety hazard, especially if there are small children in the house.
This does assume the air from the heater is evenly mixed around the room, of course. This is not always true, if for example you have a small heater in a large room. In this case you may want to turn the thermostat to maximum until the corners of the room are broadly warm, and the centre of the room is too hot. You can then reduce the thermostat to a more sensible "steady-state" temperature, and allow normal air circulation to even out the temperature.
Note that by doing this, you are technically applying "feed-forward control" - with a control system which doesn't quite deliver what you want, you're tweaking the control system's setpoint now to allow for what the system should be doing in future.
In addition to answers, I simulated an example setup to show the differences. In your case, only the referance value of the controller is changing.
I have trouble with imgur, so I also posted images online.