SSL certificate renewal - What happens to the old certificate during renewal?

Question

We have an existing SSL certificate for one of our domains which will expire next week. We know that actually there is no SSL "renewal" but we will get a new certificate instead, but we got stuck with an important question in mind.

We bought the new certificate from the same company as last year, and created a new CSR code and sent them. We wanted to complete the validation by adding a CNAME DNS record, but since there is already a CNAME record with the same name which we added last year for the validation of the existing SSL certificate, we have to actually modify the existig record.

Here our question is; will the existing old certificate be cancelled as soon as we change the CNAME record and they issue the new one? Because if that is the case, our website won't be reachable until we get the new certificate and set it up in the server, which will likely take up to an hour.

Thanks in advance.