The sensor noise you are asking about is temperature induced and not the type induced by the signal amplification (using higher ISOs).
Since both type of cameras are designed differently and technology has advanced quite a lot, both sensors are designed to have non o extremely low noise with when working within a wide range of temperatures.
In a photographic camera with a mirror sensor (SLR) is only exposed to light when while the electromechanical shutter plane opens and close at max. resolution to take a shoot, so the dissipation of the sensor is not generally so optimized.
Mirrorless cameras have the sensor exposed to light during all the time you have the camera on, and without a lens cap. But it is not taking pictures to do it so it uses a much lower resolution than when you take a shoot and the sensor is designed with more optimized dissipation at those resolutions to maintain its temperature well below it maximum operating temperature. This is the mode used while recording 1080p or 4K video.
But when you take a shoot, the mirrorless will enter in full resolution mode (stressing the sensor like the ones in DSLR) so the heat noise from each system, ceteris paribus, will be the same or even better in when using a premium mirrorless to shoot relatively very long exposure/s.
The exposure will be made, depending on the quality of the camera and the exposure speed, by clearing the sensor just before an electromechanical shutter plane opens and closes or if using extremely high exposure speeds will just reset and read the sensor extremely fast.
Lower quality cameras will just use the latter method because they don’t have electromechanical shutter planes.
As a matter of fact, today most full frame sensors (on any brand) are developed and manufactured by just one company (Sony), who is considered by the industry to bethe best and most advanced developer and maker of FF sensors in the world.
Sony’s highest quality cameras are all mirrorless. So don’t be afraid, you are in good company.