Wow as far back as 1976 the Watchtower stated swearing on the Bible is a matter for conscience. THis clearly did not filter through to my JW family.
It goes back much further than that.
"There is no Scriptural objection to taking an oath to testify to the truth. This is called being "sworn in" as a witness. It is an agreement to tell the truth on matters that the court is entitled to know."--Defending and Legally Establishing the Good News, published by Watchtower, 1950, p. 19
I've heard old-timers talk about how a JW should take great care about when and where they let themselves be put under oath, including before courts of law. I think because of misunderstanding and presumption there crept into JW lore a notion that it was somehow wrong to swear an oath with hand on the bible. The misunderstanding was that JW listeners did not understand the speaker was talking about the Theocratic War Strategy doctrine. The presumption was the oath was the typical oath with hand on the bible. But those old-timers knew they were talking about how JWs had to take care about oaths to tell the truth because when before a court once such an oath was given then the JW no longer had scriptural basis to deceive but, instead, had to tell the truth IF they opted to answer. Problem was, when before courts of law opting to not answer would leave a JW in contempt and probably incarcerated. Hence the JW had to take great care when and where they let themselves be put under oath, including before courts of law. Which is precisely what the old-timers said.