Are patents really the barrier here? Are there factories out there, in countries that aren't currently producing vaccine, and are unable to purchase it? I seriously doubt it. Making these vaccines is not trivial. The facilities required can't be whipped up overnight. The skilled staff can't be pulled out of thin air.
Most countries capable of producing vaccines like this are already doing so, as fast as possible, and exporting them to the rest of the world at an incredible rate. Sure, more and faster would be better. Sure, countries are prioritizing their own citizens. But giving away all the IP is not going to change much at all.
By the time the few other countries who are even capable of manufacturing vaccines are able to bring production online, there's likely to already be a glut of extra doses on the market. The US has vaccinated nearly half of adults. Most of those remaining are voluntarily avoiding it. Vaccine shortage isn't a problem in the US at this point. US manufacturers are still increasing output, and soon there will be tens of millions of doses being exported every month, far more than any currently-nonexistent manufacturing facility could hope to produce within the next year.
Two of the worst-off countries that could most-benefit from more vaccine are India and Brazil. India has the means, doesn't really care about IP much in the first place, and already has facilities licensed to produce, but completely failed to plan. They can solve this problem if they get their shit together; the IP isn't a barrier whatsoever. Brazil has an even bigger failure of leadership and failure of public will, and already isn't taking advantage of vaccine that they have, so again, IP isn't the problem.
The simple truth is that anyone who can make these vaccines already is. Look what happened at the Emergent plant in Maryland when they tried to make the J&J vaccine. Even in the US, with good existing facilities and--at least on paper--qualified staff, production was a failure, and many months later, it still isn't producing any vaccine.