I would love to know why one of your experts thinks this merits you leaving BT altogether.
If your email account has been hacked, moving to another Internet Provider will not stop you from getting spam/phishing or any other unwanted contact sent to that email account. Nor will it stop your contacts from getting email purporting to be coming from you if that is happening.
When an account is hacked the hackers steal the contacts that are listed in that account and possibly any stored emails that can assist them to steal your identity in order to gain access to other Internet based accounts.
They can then send phishing emails to the contacts and they spoof the hacked accounts email address to make it look as though the emails are coming from a trusted source so that the recipients feel that the email is genuine.
The hackers can also use the stolen contacts email addresses in a similar manner and in turn send you phishing emails from what you perceive to be from a trusted source in your contact list.
See link about spoofing.
https://lifehacker.com/how-spammers-spoof-your-email-address-and-how-to-prote-1579478914
If your email account is hacked you should change all security details of all Internet based accounts but.. and this is important, before doing that you must ensure that you run a malware/virus scan on your computer. After that you must check that no forwarding of emails has been set up on the hacked email account and then you must ensure that your password on that email account is changed. Once that has been done change all the other passwords of any Internet accounts.
If you are not happy using that email account, simply stop using it.
Inform all your contacts that you are no longer using it and if they receive any emails from it they should treat them as spam and delete them.
You can either set up a new BT email address of if you prefer you could set up a gmail account or other free email account. If you don't mind paying for your email account you could set up your own domain with email for a few pounds a month. There are many to choose from.
If you presently use the BT email address with any Internet based accounts such as banks etc you would need to inform them and give them your new email address.
As regards the email you have received. If you are not the BT Broadband account holder why have they contacted you?
It may well be that the email you have received is a phishing email. You should never click on any links in order to respond to a request to change passwords etc. You could be going to a fake web site and end up entering your user name and password into a scammers fake web site and you will just have given them they keys to the kingdom.
If you have to do that you should go directly to the relevant web site and change the details from there.
The other thing that may have happened and in my opinion is the most likely due to the number of other reporting having received a similar message is that BT have restarted the migration of BTYahoo email accounts to the BTMail system.
It may be that your email accounts are part of the migration process and as a result the "system" has seen this as unusual activity and the generic message was sent out.
