All things being equal, is a 1 Inch 20MP sensor basically the same as a 1/2.3" 12MP sensor?
Short answer : not the same. There's a significant difference.
Which sensor size is better depends on your needs.
I have an RX 100 Mk3 and it has a 1" sensor. That model has a lens with a relatively wide aperture so the combination does quite well in low light (but it also has a flash !).
I've considered the 1/2.3" Panasonic P&S models including the one your mention. They do provide more focal length range, which is something the small sensor allows in a compact that is not at all easy in a large sensor model.
As far as light gather goes, the smaller sensor ( it is approximately half the size in length and width and hence roughly one quarter the area of the 1" sensor ) it is not able to gather as much light. For design reasons the zoom range means you also have a narrower aperture so it gathers less light to start with. It's probably about a stop to two stops less capable.
They simply do different things.
Asking specifically about low-light and image quality. Zoom is not the issue here.
Consider the combination of aperture values and sensor size light gathering and make your own mind up.
But the sacrifice of focal length is significant. It does frustrate me sometimes that I'd stuck with a 24-70 equivalent and can't get the extra focal length range.
If you're particularly chasing low light performance I'd suggest a larger sensor, like the ones in the Sony NEX and A6xxx models with a compact kit 16-50 lens. These also give you the option of mounting a "proper" fast lens if you're willing to add some weight and bulk.