From what I see on these forums, BT have been having problems with email for years. IIRC in the time I've been here they have changed the supplier three times, for reasons I don't know but could be profit or because they can't find one that works.
By lunchtime today you could have a domain of your own with web space and working email up and running for £22.18 per year. That's equivalent to £1.85pm. More info and a link to my current and recommended supplier on this page
Their basic service and helpline are excellent, 24/7. I have had one failure of their email and no failure of websites since switching to them in 2010. I rarely need their help but when I do they reply with spot-on help within minutes.
If you go for this solution with them, opt for the cPanel hosting not the Cloud one. You get more control in the future.
Get it up and running and start telling people/shops/customers/whatever your new address. Never send on BT again, just receive from legacy places you haven't told.
Kindness isn't going to cure the world of all its awfulness but it's a good place to begin. Daisy Ridley.
My broadband basic info/help site - www.robertos.me.uk.
Domains, site and mail hosting - Tsohost
Connection - AAISP Home::1 80/20. Sync 57825/13835kbps @ 600m. - BQM