Switching email providers without losing mail: the safe order
Move in this order: new account first, then forwarding, then domain DNS, then the old archive. Swapping DNS first is the mistake that loses mail.
Step 1, open the new account. Create the mailbox at the new provider and test sending and receiving for a full day. Most European providers in this directory import mail by IMAP, so leave the old account untouched for now.
Step 2, run both in parallel. Set the old provider to forward to the new address, and reply from the new one. Two weeks of parallel running catches the senders who only write twice a year, like tax offices and registries.
Step 3, move the domain. If you use your own domain, change the MX records only after steps 1 and 2 work. Keep the old provider active for at least 30 days after the MX change, because some senders cache DNS for days.
Step 4, export and close. Download a full archive in mbox format before closing the old account. Storage is cheap, regret is not. Then delete the old mailbox so the data stops sitting under the old jurisdiction.