COS Start date and real work date

Hi all

I have a question about the COS start date.
When my new employer issued me the COS, they put my work start date on 1st April.
I submitted my visa application on 24th March and used Priority service.
But the earliest biometric appointment date is 13th April.
I asked the immigration team of my current employer- an university about the two dates mentioned above. They told me my new employer should change the date on COS to reflect my real work start date.
May I ask if this is a must to do? If my employer is going to change the date, how do I need to change my visa application and biometric appointment?

thank you for your kind reply

Just added that I applied the new visa from the UK

Sorry about adding more info as I think my current visa status is important too.
My current job finished on 31st March and my current visa will expire on 14th April.
My new employer stated the job new start date on 1st April and I submitted my visa application on 24th March but the earliest appointment date I got is 13th April.
In this case, will my new employer need to change the COS? if they change it, do I need to change my visa application?
Due to the Easter holiday and the unpredictable process time, if I change the application after 14th April (my visa expiry date), am I still able to stay legally?

thank you so much

Just an update for my question as I called UKVI for my question. They said I don't need to make changes on my part. The employer can change or no need to change, no influence on the visa decision.