Your posting gives me pause. I just purchased the exact same phones (last night) and have a very similar DSL set-up. I haven't connected the Uniden system yet because I guess I'm supposed to let the phones charge overnight first, but I was planning to do so tonight.
I did look through the manual briefly this morning and I noted there are two types of voice mail service signals and one type may require a configuration change on the Unidens. It may not be the culprit, but I thought I would throw it out there as an idea.
If you do find the culprit, please post.