When it’s nearer your completion date you will start getting notifications that your direct debits have moved over. This will help reassure you that everything is on track
Yep, CASS service takes care of it. One of my DDs didn’t update in time, as the account switch was very close to the date the DD was due. Even though the company sent the direct debit request to the old (closed) account it was still redirected automatically and debited from my Monzo account. By the next month it had been updated with the company and the request went directly to Monzo for payment.