CoS Start Date

Hi,

Is it ok for the start date of my new CoS to be the day my current leave to remain expires? i.e. if my current leave to remain expires on 5th May, my employer will issue the new CoS has starting 5th of May.

this is fine.
although if this is an extension application, you should try putting the next date from the end of your previous cos.
also, the end date of your visa will not be the end date of your previous cos. ususally, the visa is given for 14 days plus cos end date.

Thank you for your advice: we will apply for my new CoS to start the day after my current CoS expires.

So, if my new CoS starts tomorrow, my current leave to remain expires in two weeks. Is it fine for me to apply for my visa extension within those two weeks, after the new CoS has been assigned and much closer to the date my current leave to remain expires?

you could/should apply any time before your current leave expires.