Start/end date of new CoS vs. current visa

[Tosh119]

I am currently on a tier 2 visa that expires in November this year. I am switching companies, and my new CoS has a start date of 5/4/2021. I have not yet given notice yet, as I haven't applied for my visa yet, and we haven't confirmed the start date. Will I still be able to work my current job after the 5th of April if I cannot start the new one then or do I have to stop working by that date?

Also, does anyone know if the end date of the new CoS can be amended as a sponsor note? The date they've got on it now would (I think) put me a few weeks shy of 5 years for ILR because I think we were all a bit unclear on what date UKVI uses for the qualifying period (e.g. start date of original CoS, entry clearance valid date on passport, or date on biometric permit).

[CR001]

There is nothing unclear about the ILR qualifying start date. It is the initial visa issue date.