Keir Starmer (or for that matter Theresa May and David Cameron too) would have included us in the lame duck pan-European vaccine procurement strategy. We wouldn't be in the strong position we are in now.
The EU procurement strategy was completely up to each country. Even if we had been involved, we might have been able to help to speed up the process if we were keen for the vaccines earlier, as we could have been involved in the talks on that.
I highly doubt that the first lockdown would have come any sooner under a different PM. Remember that the medical/scientific advisors would be the same people and that they were worried that a strict lockdown might be counter-productive.
The Chief Medical Officer is appointed by the government, with an external process, but the gov website says "The appointment has been made with the support of the Prime Minister". Plus meetings can be held with certain slants on what is going to happen. If Johnson had wanted to take a Jacinta Ardern approach, he could have asked the medical experts on their advice on how to go for a zero covid strategy instead. Instead of that, he didn't bother even going to the first few COBRA meetings to find out what was going on.
Labour might be disappointing, lacklustre and ineffective at the moment, but they would have been extremely unlikely to have been up for letting the virus 'sweep through' or for bodies to pile up. They would have been extremely unlikely to give vital jobs like PPE procurement or ventilator manufacture to their mates in completely unrelated fields of work. The mainstream media is predominantly against them, so they would have had way more eyes on what they were doing, too.
Probably if they'd been in, the media would have been criticising Starmer for 20,000 dead or something, unaware of how terrible this whole timeline would have been instead.