The shock move, on the eve of the new campaign, comes just months after the 38-year-old American spent time on the summer league staff of the Dallas Mavericks.
The decision is not expected to affect Finch's commitments with the GB side but will see the ex-Sheffield boss depart Mons, who he took to a European final, at the end of this week.
“Chris is a young, up-and-coming coach who already has a wealth of coaching experience,” said Rockets General Manager Daryl Morey who, according to reports, had a major role in the appointment.
"Chris has been instrumental in the re-building of Great Britain's fledgling national team program and I believe he will employ that same passion and commitment to elevating the level of Vipers basketball. We will work closely with Chris to ensure that the Vipers implement the same successful philosophies employed by the Rockets."
Nurse's assistant on the British team, Nick Nurse, is coach and GM of D-League rivals, Iowa Energy.