Tier 2 Application (Change of Employment) - Help

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Hi, I'm currently on a Tier 2 visa (Tier 4 to Tier 2 switch) valid till 18/11/2021. i've been offered a job with an employer who didn't have a sponsorship license but applied for one and they received it this friday 20/11/2020. The thing is they obtained the sponsorship licence but don't have any allocated unrestricted COS available as they didn't provide the evidence about the migrant.

So I've provided them all of my details and they will make a COS application via the SMS now.

I've just got a few questions regarding this.
1.Would the new COS application with the new future employer affect my current visa status with my current employer?
2.What is the timeline for COS applications through the SMS system?
3. My notice period is 30 day with the current employer, when would be the ideal time to hand-in my notice?
4.Can i make a make a fresh Tier 2 change of employment application whilst still being employed with my current employer and only hand in my notice with i receive the new BRP? In this way i won't have to worry about the curtailment of my visa.

Thanks.

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I would be very grateful for any sort of help here. Thanks!