Thank you for the comprehensive citations in your summary on the page. It makes it easy to follow.
One you may have missed is this one Wiltshire Archaeology and History Magazine 1949 which follows one cited in the profile.
He mentions that Oglander, who was writing in 1626 mentions that Sir Nicholas (who was Captain of the Isle of Wight from 1509-20) married a sister of Jane Seymour and that both were buried at Carisbrooke. That may have been somewhat garbled (ie Nicholas certainly not buried on the IOW and Margaret may not have been the sister of Jane Seymour )but Oglander also mentions that Sir Nicholas seemed to have fallen out of love with the Island and sold land there after her death https://archive.org/stream/oglandermemoirs00oglagoog#page/n140/mode/2up
Oglander seems to be the earliest reference I could find.
The older County Historians also include a marriage between Nicholas and a Margaret Seymour . Collinson,1757-93, Somerset, says sister of Jane. Albin, 1795 IOW daughter of sir John and sister of Jane) Hutchins, Hist and Antiq Dorset Vol 2, page 216, Wadham pedigree) states that Nicholas married a sister of Sir John Seymour, he also gives no evidence but mentions a record granted to Nicholas and Isabel (his 3rd wife) the lands of her late husband. This was dated 1518.which gives a ball park date for his previous wife's death.
I think the two memorials, if the arms on them have been interpreted correctly (we need an 'expert') add very strong support to his marriage to a Seymour since both apparently have the arms of Wadham impaling Seymour
1) The infant son buried at Ilton, Somerset https://commons.wikimedia.org/wiki/File:NicholasWadham_1508Brass_IltonChurch_Somerset.JPG
2) The uninscribed tomb at Carisbrooke https://commons.wikimedia.org/wiki/File:MargaretSeymourWadham_CarisbrookeChurch_IsleOfWight.PNG (the VCH certainly seems to accept this to be the tomb of Margaret (footnotes cites Percy Stone Archi Antiq of Isle of Wight and states she was the aunt of Jane Seymour)
The Visitations of Somerset also has Nicholas marrying a sister of Sir John Seymer https://archive.org/stream/visitationsofcou00beno#page/84/mode/2up
So all this suggests that Nicholas was married to a Margaret Seymour (on balance, I think aunt )
But what about Sir William Wadham apparently also married to Margaret Seymour; same Margaret, different one ? The profile doesn't help by having no dates.
Who was Sir William?
It can't be https://en.wikipedia.org/wiki/William_Wadham_(died_1452) who seems to have had two wives, well accounted for.
Unfortunately, the Richardson source used is not accessible (nearest library that has it according to World Cat is3,100 miles away in Boston) I have to say this is a great problem for us in the UK so page number etc doesn't help much. If it is considered to be an important secondary source, I wondered why there seem to be no copies in the major copyright libraries and then realised that it was self published...
What we need to know is which William Wadham ie his parentage and dates. Which Margaret, her parentage and dates and what Richardson uses as evidence. This could be summarized on his profile without infringing copyright