"But the AstraZeneca vaccine, in particular, and the Pfizer vaccine are going very well."
In New Zealand, PM Jacinda Ardern has said the rollout for a vaccination would likely be in March.
The UK began rolling out the Pfizer-BioNTech vaccine earlier this week. But British authorities have cautioned people with severe allergies against taking the shot, after two patients had reactions on the first day of the rollout.
Despite its approval for rollout in the UK, the Pfizer vaccine will continue to undergo trials, a fact that has sparked concerns over unforeseen side effects.
The UK has recorded more than 10,000 cases every day this week and more than 60,000 Britons have died from Covid-19.
Morrison said the dire situation meant the UK had "no choice" but to expedite its rollout, but downplayed fears Australia would face similar hiccups.
"Australia will have a front-row seat to how that goes, and we'll learn from that," he said.
"But the health side of this is paramount. The Therapeutic Goods Administration must give its tick-off, so there are no shortcuts there."
He confirmed he would discuss the vaccine rollout schedule with state and territory leaders when National Cabinet met on Friday.
It comes after Health Minister Greg Hunt claimed last week the federal government was "ahead of schedule" in its five-step vaccination plan.
He told reporters he was hopeful Australia would be in a position to approve at least once vaccine by January or February.
