COG1 acknowledgment

I sent in a change of address form early December. Should I expect an acknowledgment when the register is updated?
I am signed up for the alert service, and did receive an alert (see details below), so I know that the form has been received and logged. The alert just said an application has been lodged though. It doesn't actually say the register has been updated. With no  official acknowledgment, can I be sure that the change has been made to the register?
Thanks
Phil
Property Alert reference: 4047512
On: 06.Dec.2023 At: 11:44:27
Customer reference: NONE
HM Land Registry reference: J099TFT

Phil - once processed we will write to confirm the update and issue a fresh copy of the register. You just need to wait to hear from us

OK. Thanks. Hopefully, you'll write to the new address.
Will it also generate another alert?

No - the alert is simply to notify you of receipt of a significant application. Not it's completion.