The Rush Maxwell Blodget Codex (v1 p134, 135) lists a transcript of a marriage record from Suffolk, citing "Boyd's index of Suffolk Marriages, at Ipswich Public Library":
"1561 Robert Blogat[e] m. Margaret Waren at Bacton"
There's a Bacton in Suffolk about 8km from Stowmarket Haughley, so it does seem like a fine candidate for this Blodgett/Blogatt/Blogate family.
The transcript of Robert's will (dated 5 Jul 1602, proved 2 Aug 1602) names "To Marie my wife £30..."
Presumably it is on these bases that the internet has assigned Robert's wife the name "Margaret Marie". The names are similar, but barring other evidence I'd also wonder whether they might be two different wives of Robert.
Vincent Redstone sent some transcribed burial records from Stowmarket Haughley, in which there's no wife of Robert listed, by either name.
I don't see any other references to her in the Codex materials that I have, so it looks like perhaps Dr. Blodget did not commission any further research into the Waren origin in Bacton. I don't know where the parentage claim may have come from; I'll keep looking.
Updates: there's a link on the profile of John Warren to a website with a transcript of his will, dated 1576. No daughter Margaret is mentioned. Even if Margaret was a first wife and died before 1576, I might expect to see a bequest to her children, John's grandchildren, in his will. Not definitive, of course. She/they might have been legitimately omitted for a number of reasons.
Also, the Warren family is of Nayland and doesn't seem to have any connection to Bacton. They're about 30 miles apart, both in Suffolk.
More Updates: FindMyPast.co.uk (I don't have a subscription so can't see detail) has three listings for the name Waren/Warren at Bacton:
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Margaret Waren b. or bap. 1539, which was exciting until I saw also d. or bur. 1539
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Agnes Waren d. or bur. 1540
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Elizabeth Warren d. or bur. 1558
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Thomas Waren d. or bur. 1566
There are listings for Warren/Warreyn in Haughley starting about 100 years later, none contemporaneous.
FamilySearch.org/LDS.org have digital copies of Boyd's Index, which lists these Waren/Warren marriages at Bacton:
And this is the only one I found in Haughley or Stowmarket:
No idea how they might be connected. Cheers :)