The UK could've done exactly what it did whilst an EU member.
Why no other countries did it? Who knows. Maybe they thought they were already paying into the EU-wide scheme so why bother?
1: https://en.wikipedia.org/wiki/COVID-19_vaccination_in_Hungar...
https://www.bbc.co.uk/news/explainers-52380823
> Hungary has been using Russia's Sputnik vaccine for months and and Slovakia has also bought doses of it.
> Germany said in April it would begin talks to secure supplies of the Russian vaccine and Austria said it would buy one million doses, but only once the EU's medicines agency approves the jab for use in the EU.
> The Czech Republic and Poland have entered talks with both Russia and China.
So that's Hungary, Slovakia, Germany, the Czech Republic and Poland - that would be 5 states, ~20%.
Ironically, a non-brexited UK government opposing the single-procurement EU program would have probably forced more EU countries to go alone. On practical matters, it's much easier to sabotage anything when you're inside the building, rather than outside.
Our success at procuring vaccines was, at best, dumb luck. If we had any idea what we were doing as a country, our death rate wouldn't have been so high and perhaps we wouldn't have needed to be so desperate for vaccines.
It's like setting yourself on fire and then being really proud of yourself when you manage to find a fire extinguisher.