The Olympic arrived in Belfast on 6th October, 1912 after being patched up in Southampton, which took two weeks. On 20th November after several weeks of repairs, she left Belfast returning to commerical service.
The Olympic was repaired very quickly indeed, but we must all remember that the quick turnaround back into service was due to the fact that H&W nicked the already machined replacement parts, i.e. the starboard boss shaft and propeller, from Titanic - so in saving Olympic weeks in drydock they instead added weeks to the completion date of Titanic which set back her Maiden Voyage from March to April.
I've since found out that the Olympic resumed her voyages on November 29th - the day before Britannic's keel was laid. This must have been good publicity for the White Star Line.
On a different note, how soon after the Hawke collision did WSL advertise the Olympic's new sailing schedule? Did "they" say "Take as much time as you want to repair the ship" or "you have x number of weeks"?
The quick repair time of the Olympic and coincidence that it was completed the day before the keel of Britannic was laid was hardly good publicity for White Star.
The celebrated inquiry into the incident lasted well into two months, and after hearing testimony from all involved, decided in Hawke's favor finding Olympic at fault. White Star's appeal of the decision encouraged a firestorm of negative publicity and controversy.
Even considering that the Olympic was under the command of a Trinity House pilot and not White Star officers at the time of the collision, White Star was still held liable and ordered to pay exorbitant legal fees in addition to the cost of repairing the Olympic.
So to answer your question, I think it was in the best interest of both WSL and H&W to get the O back in service as quick as possible to recoup the millions of dollars in lost operational income as quickly as possible.