What happened to "proceed" and "precede"?

When proceed entered English in the late 14th century, it was spelled similarly to Latin and French: procede, and until the 18th century, this spelling competed with the original. Precede is first attested somewhat later, but the spelling preceed soon arose, and to judge by the results in a Google Books query, there are still modern writers confused by the spelling. In the 16th and 17th centuries, there was no recognized authority capable of regulating the chaos of English orthography, so authors were free to choose any spelling, which might vary even within one work.

It is fit that the Lords of Session, who have but a Temporary Precedency, should not preceed them. — Sir George Mackenzie, Observations Upon the Laws and Customs of Nations, as to Precedency, 1680.

And warn the foresaid S. that then he be there to preceed in the plea foresaid, and to doe and receive what our court shall consider in the premises. — Anthony Fitzherbert, The New Natura Brevium of the Most Reverend Judge, Mr. Anthony ..., 1652.

…and howe I should in loue procede. — John Stow, ed., Woorkes of Geffrey Chaucer, 1561

…we now procede to the bodye of Antichriſtianiſme… — Theophilus Gale, The Court of the Gentiles, 1672.

By the time this chaos reaches Samuel Johnson's dictionary of 1755 it has become ordered into the modern — but often still confused — spellings that dictionaries prescribe. Both precede and proceed are consistently spelled as today, but in the entry for celestial, Johnson duly notes Pope‘s alternate spelling in an illustrative verse:

Thus affable and mild, the prince procedes,
And to the dome th' unknown celestial leads. Pope's Odyssey.

After Johnson, alternative spellings become quite rare, mostly appearing in citations from older works, or, of course, from the French. Whether this normalization into contrasting spellings was due to Johnson's influence or trends in orthography to which his dictionary is a witness, it's pointless to search for a specific reason why precede won out over preceed or proceed over procede — even more remarkable with the word procedure floating about. Orthography, like virtually all features of language, is a matter of one writer or speaker simply imitating another, and these spellings are the result.

I doubt a complete explanation can be had.
But, it is clear that the spelling of precede was more or less codified before the final syllable "collapsed".
Sometime between 1500 and 1700 a great many words lost their final syllable when the final e after a consonant stopped being pronounced. It is generally thought The Great Vowel Shift precipitated the sound changes that brought this on. There is no question sound changes happened, but there is only reasonable guessing as to why syllables were lost.

Proceed was, and is, spelt differently.That could mean next to nothing, as there were no dictionaries to consult, or, that the root "ceed" was pronounced differently from the "cede" in precede. But, there is no solid reason to think that. The quality of the vowel in the prefix might have influenced the pronunciation of the root, but, again, there is no hard evidence.

Another explanation might be that proceed did not gain enough currency to have a codified spelling until after The Great Vowel Shift had ended; that is, ended where printing was done. There would have been no reason to spell a word with three syllables that had only two, not that English always "makes sense".
A lot of words got caught in the advance of mass printing. Spelling was sort of "mummified" even as the sound of words was changing. The sounds evolved in different places at different times, and in different ways. There are some things about English that reason and logic will not explain.