Indeed, the standard MT has Jehoiakim. The BSB and many modern versions have "Zedekiah". The BSB has this footnote:
A few Hebrew manuscripts and Syriac (see also verses 3 and 12, and
Jeremiah 28:1); most Hebrew manuscripts Jehoiakim
Ellicott expresses a possible explanation:
(1) In the beginning of the reign of Jehoiakim.—The mention of the
name of Zedekiah as king of Judah in Jeremiah 27:3 shows that the
Hebrew text has here perpetuated an error, due probably to the
transcriber or first editor of the collected prophecies. We have to
think, accordingly, of the state of things which followed on the death
of Jehoiakim, and the deposition and exile of Jehoiachin. The tone of
the prophecy seems to indicate a time about the middle of Zedekiah’s
reign. His position was that of a tributary sovereign, subject to
Nebuchadnezzar. He and the neighbouring kings, who were in a like
position, had not quite renounced the hope of throwing off the yoke,
and asserting their independence.
Benson makes similar comments:
Jeremiah 27:1. In the beginning of the reign of Jehoiakim — Instead of
Jehoiakim here, Dr. Waterland, Houbigant, Blaney, and many others,
read Zedekiah, because it is difficult, if not impossible, to
reconcile the common reading with what follows. Lowth also, in his
commentary upon the place, gives it as his opinion, that “the least
forced way of solving the difficulty is, to say that Jehoiakim has
crept into the text by the negligence of the scribes, (who might have
their eyes fixed upon the beginning of the last chapter or section,)
instead of Zedekiah. This emendation is confirmed by comparing this
verse with the 3d, 12th, and 20th verses of this chapter, and with the
beginning of the next. Such little verbal mistakes must be allowed by
all impartial readers to have sometimes happened in transcribing the
Holy Scriptures, as well as in other books, and may easily be
corrected, by comparing the suspected reading with other parts of the
sacred text, which admit of no difficulty or uncertainty.”
Other commentators offer similar remarks.