John Gill was a Baptist preacher in England who became a pastor in 1719 and died in 1771. ... Not sure how representative Gill is of all Baptists in England in his day, but this at least trace the "no divorce" idea to 1746, when he published his Exposition of the Whole New Testament.
You might go back earlier. The Westminster Confession (Presbyterian, 1646) set out the grounds for divorce:
" ... In the case of adultery after marriage, it is lawful for the innocent party to sue out a divorce, and after the divorce to marry another, as if the offending party were dead. Although the corruption of man be such as is apt to study arguments, unduly to put asunder those whom God hath joined together in marriage; yet nothing but adultery, or such willful desertion as can no way be remedied by the Church or civil magistrate, is cause sufficient of dissolving the bond of marriage; wherein a public and orderly course of proceeding is to be observed; and the persons concerned in it, not left to their own wills and discretion in their own case."
The Westminster Confession was a major source for the Second London Baptist Confession (1677/1689), yet the London Confession (following the Independent Savoy Declaration of 1658, which the Baptist confession follows more closely than the Westminster) omitted the section on divorce. Whether this reflects a genuine disagreement with the Westminster Assembly, I cannot say. Parliament approved the Westminster Confession minus some sections, including the one on divorce. Did the Independents (and thus the Baptists) ignore the section because they disagreed or because they were unwilling to tread where Parliament had not gone? I am not aware of a definitive answer.
I might also hazard a guess that divorce was not a pressing issue with the 17 century English Baptists. Divorce (which required an act of Parliament) was expensive and a long-drawn-out affair (so to speak) and very few of those Baptists would have the means to seek a divorce even if they desired one.
I would pass on this letter from John Broadus in 1885:
"The case you describe belongs to a class sometimes causing great pain in the action necessary to be taken. To me it seems perfectly plain that our Lord expressly prohibits divorce, except for the cause of unchastity. When, therefore, two persons have been by law divorced on other grounds, and one of them, during the lifetime of the other, marries again, I do not see how a church can recognize this second marriage, or fail to treat it as adulterous. It may sometimes be a very painful action to take."
