Assombalonga soon equalised with a simple close-range finish, but Wheater re-established the home side's lead moments later by poking home at the near post.
The former Peterborough United hitman then converted a penalty after being fouled by Spearing in the area to cap an end-to-end first-half.
Both sides maintained the frenetic tempo after the break and the pendulum very nearly swung in Bolton's direction when a glancing header from Craig Davies went just wide on 53 minutes.
Assombalonga could, and perhaps should, have bagged his first hat-trick in England's second tier soon after when he headed a Michail Antonio cross narrowly wide of the post.
The two clubs cancelled each other out for the majority of the second half, although Davies again had a good chance to nod Bolton into the lead when his 83rd-minute header sailed over the bar.
It proved the last opportunity of the game as both teams settled for a share of the spoils.