As long as we haven't lost the sun too, we can sort it all out again reasonably quickly.

Now you have introduced an additional variable (location) to the equation.

Especially as mechanical watches drift a few minutes each day.

A few seconds each day, unless you have a really bad or poorly maintained watch.

It must be exceptionally bad mechanical watches you are used to

The drift is affected by movement right? I imagine sitting on a table they are much more accurate than swinging around on an arm all day.

Perhaps, but they're engineered to minimize such effect. I noted in another comment I started with a Seiko 5, one of the cheapest, most inaccurate movements (being the name for the timekeeping mechanism itself), and I see multi-minute differences after weeks without adjustment. I'm not a daily runner or anything (in which I'd use my smartwatch anyways), but my job does involve a fair amount of getting up and walking between places, as well as the temperature shift of controlled environment rooms for computers and SoCal outdoor heat.