Two simple questions arise: (1) who decided to put the two verses together? and (2) why was this done?
The second question turns out to be more easy than the first: compared to those older editions, the verses would actually seem to have been put back together. The very first edition with the still current system of New Testament verse numbers, Robertus Stephanus’s 1551 diglot, had 51 verses in John 1, just as today, and John 1:38-51 exactly divided as it is now.
This observation raises two additional questions: (3) when was verse 38 first split in two? and (4) why was this done?
To start with the third question: Stephanus’s 1555 Latin Bible, and Beza’s first major New Testament edition of 1556/1557 did not split the verse. The first edition with 52 verses in John 1 that I was able to find is Beza’s second major edition of 1565. It is hard to say whether the change reflects a decision by the typesetters or by Beza. In any case, the split verse on p. 362 (of part 1) is not an error or accident. Indeed, just as all the other verses, verses 38 and 39 are formatted as separate paragraphs; moreover, Beza’s annotations to John 1:34-44, printed on the same page, follow the new numbering.
It is very likely that this 1565 edition was indeed the first with the split verse, for surprisingly enough the remaining portion of John 1 on its next page (p. 363) is still numbered as verses 44 to 51, instead of 45 to 52, and the annotations again follow suit.
Beza’s 1565 minor edition has the new numbering of John 1, but now without flaws or hesitations, and so it remains in Beza’s editions, albeit with some oddities here and there. The fact that Greek New Testaments such as the famous Elzevir editions follow Beza’s numbering may be an additional indication that these editions are closer to Beza than to Stephanus.
A reason for Beza or his typesetters to change the numbering, my fourth question, is not easily found. The old verse 38 may seem rather long, but its length is not excessive. In any case, names can now be attached to the two systems: a version of John 1 with 51 verses follows Stephanic numbering, whereas it having 52 verses points to Bezan numbering.
Apparently then, these two numbering systems happily (?) coexisted for some centuries. It seems a case of England against the continent, but not entirely so. Their coexistence finally leads to the answers to my first question: why is the Stephanic numbering of John 1 the only one used today? It turns out it was chosen by Eberhard Nestle for his important 1898 edition, which he based on those of Westcott-Hort, Weymouth, Tischendorf, and Weiss. Confronted with the divergences in numbering, he opted expressly for the oldest, original one:
Verses … are numbered as found in Westcott-Hort and Weymouth, according to the 1550 [sic] Stephanus edition in which they were first introduced; from which regrettably later editions, among which Tischendorf and Weiss, sometimes diverge.Indeed, a large amount of confusion, as well as this small contribution, would have been unnecessary, had not Beza’s 1565 edition introduced an extra verse in John’s first chapter.
In sum: (a) in 1565 Beza introduced an extra verse in John 1, by splitting verse 38 in two; (b) for centuries, two differently numbered versions of John 1 coexisted; (c) in 1898 Eberhard Nestle took the influential decision to adopt the original, Stephanic numbering.
 See below for Acts 20. There must be many more cases, isolated to a few editions only. E.g. the Dutch 1562 Deux-Aes Bible, followed by the 1637 Statenvertaling, splits Rom 7:25 in two; Whittingham NT 1557 and GB 1560, followed by KJV 1611, split Acts 19:40 in two. KJV 1611 splits 2 Cor 13:12 and thus has a verse 14 in this chapter (I guess because the split coincides with the beginning of a new column).
 There is however an error in verse numbering on p. 280v: verses 26-30 are numbered 27-31, and there is no verse 26; the first verse on the following page is again 31. Of course this error has no bearing on the rest of the chapter.
 The 1565 minor edition is later than the 1565 major edition: the former’s dedicatory letter to Louis I de Bourbon is dated “x. Calendas Martias” (February 20th) of 1565 (p. ¶¶ iiiiv), whereas the latter’s dedicatory letter to Queen Elizabeth is dated earlier, namely December 19th, 1564 (p. * vv).
 Such as a missing verse number for the annotation on verse 40 in the 1582 major edition; or, in the same edition, verse 52 suddenly numbered as 51.
 Stephanic (51) numbering of John 1 is found in (just a selection): Stephanus NT 1551; Beza NT 1556; Whittingham NT 1557; Geneva Bible 1560; Bieskensbijbel 1560; Deux-Aesbijbel 1562; Bishop’s Bible 1568; Rheims NT 1582; Hutter NT 1559 and 1602; KJV 1611; Amelote NT 1686; Simon NT 1702; Tregelles GNT; Westcott-Hort NT 11881; Nestle NTG 11898.
Bezan (52) numbering is found in (again just a selection): Beza NT 21565 31582 41589 51598; Elzevir NTG 11624 21633; Statenvertaling 1637; Whittaker GNT 1633; Curcellaeus NT 1658; Schmidt Versio 1658 (though John 1:38-39 is printed as a single paragraph); Fell GNT 1675; Mill NTG 1707; Wettstein NTG 1751; Harwood NT 1768; Griesbach NTG 11777; Lachmann NTG 11831; Tischendorf NTG 81869; Baljon NTG 1898.
 “Versus … Numerantur ut apud HR [Westcott-Hort and Weymouth] secundum editionem Stephanicam anni 1550 [sic], i. e. primam, in quam introducti sunt; dolendum, posteriores, TW [Tischendorf and Weiss] quoque, interdum ab illa recessisse” (p. 660; my translation).
 Or to put it differently: … had the entire world adopted Beza’s numbering. Indeed, in Acts 24, the same 1565 edition takes verse 18 and 19 together, thus producing a chapter with 27 verses instead of the 28 verses found in Stephanus’s 1551 edition and Beza’s own 1556/1557 edition. In this case, the Bezan numbering won the day and the Stephanic numbering quickly disappeared from the scene. It is perhaps a matter of luck that Nestle was not aware of the Stephanic numbering at this point …