Mominul Haque was on 64 while allrounder Shakib Al Hasan was 5 not out when umpires Paul Reiffel and Marais Erasmus took the players off at 1830 (0530 GMT) due to bad light and abandoned play 15 minutes later.
The players had left the field before lunch and tea due to persistent rain and returned during the third session only to be forced off before the umpires even reached the middle.
They returned for a final time about an hour later but bad light ended play after a frustrating 40.2 overs.
The showers, powered along by gale-force northerly winds, forced an early lunch after 11.3 overs with the visitors on 39-1, having lost opener Imrul Kayes for one, caught by Boult at deep fine-leg after he hooked a Tim Southee bouncer.
He then hit his 11th boundary but was struck in front by Boult off the next ball and while he was initially given not out, Williamson had the decision overturned on review.
Bangladesh bowlers Taskin Ahmed and Subashis Roy were given their Test debuts, with the 21-year-old Taskin expected to lead an inexperienced pace attack.
Medium-fast bowler Kamrul Islam Rabbi has played just two previous Tests.