After nearly 5 years, TTT! 
Currently looking at my 'Early Music' collection and am on William Byrd - presently if have 14 CDs (including multi-disc sets), shown below. I feel the KB & consort works are covered, along w/ the masses - but what seems to be missing are more 'vocal' pieces; also I'm curious about whether any recent 'piano' versions of these keyboard works have emerged recently. Any comments and suggestions appreciated. Thanks. Dave
Ashamed that I've posted nothing here for 5 years, too. Dave, you look to be well covered with the Masses courtesy of the Tallis Scholars. All of the Cardinall's Musick/Andrew Carwood discs are essential listening for Byrdians, but if you want to start with one, start at the end:

The Great Service is an unmissable masterpiece, of which there are plenty of recordings (again, the Cardinalls are good), but this one is a bit unusual:

- the only one, to my knowledge, performed at Byrd's written pitch rather than the standard minor 3rd higher.
A fine newcomer, which includes several previously unrecorded pieces, is:

I would second Mandryka's recommendations of the
Adoramus te and
Songs of sundrie natures discs - although the latter actually includes very little from Byrd's publication of that name. (David Skinner tells me that the '1588' disc is planned as the first of 3 to cover Byrd's three English song-books, funding permitting.)
You could make room on your shelves for some of these by ditching Elizabeth Farr - lumpen, joyless and
slow - every timing way longer than Christopher Hogwood's
Nevell set, another bit of essential listening (although of course it's all covered in Davitt Moroney's box anyway).
... the Cardinall's Musick on Hyperion. Was meaning to acquire more by that ensemble, but as noted above their "complete" series seems to have stalled out, and even [IMO] been rendered unlikely by a recording label change.
Fairly early in the project Andrew Carwood decided that "complete" was going to mean "complete Latin Sacred" - his opinion was, probably correctly, that the Cardinalls were not suited to the lighter stuff in the English songbooks. The switch from ASV to Hyperion was unfortunate, as it probably means that the whole set will never be available in a single box. We Byrdians did rejoice when the '1588' discs appeared, as it now looks as though the published English music will be recorded complete. That leaves the unpublished anthems and services for someone else to pick up...