Now I've never seen mail rejected like this before, a message sent to username@mydomain.org.uk is being rejected as username@dynamicdns.no-ip.org surely the DNS redirect should be transparent?
But that is not a DNS redirect (which might be what you could call a CNAME), but that is an SMTP recipient re-writing, which is apparently being done by the MX for mydomain.org.uk.
I've now set dynamicdns.no-ip.org as a domain on my server and mail sent to username@mydomain.org.uk gets through fine
Good, you solved that problem.
but mail sent to username@otherdomain.co.uk never appears on my server and the senders account never gets a failure message.
So you need to check the DNS of otherdomain.co.uk, and then work out what is happening at whatever mail server is listed as an MX for that domain (or A record for the domain if there is no MX record).
As for the "no failure message", some mail servers no longer bother to provide those, or if they do, the failure messages aren't being delivered. We have a lot to thank spammers for

.