FWIW I have had no problems using POP but leaving mail on the server with VM (and before that BY) . But I may just have been lucky. (And I do it on the basis of using POP to clear the mail i.d.c. because the webmail interface is slow.)
Idiot boy question on the more general problem: is it essential to share the same email addresses across the 2 machines? VM (ex-BY) offer 5 mail boxes with 3 aliases on each. So would it be acceptable to have separate mailboxes so we each PC downloads only the mail for that mailbox?
Second idiot boy question: did the new machine come with security software which might need to be configured to allow email access?