My take is that situations like this often hinge on how they are justified and how that is recorded. It's not how I would deal with it, but I'm pretty confident it would be found to be lawful. Outside of a prosecution it would be harder to justify as police officer due to the requirement of using the minimum force necessary but bouncers don't have the College of Policing Code of Ethics to abide by. With that said, whilst it is harder, I think even then you could probably justify it.
My reasoning would be that the first punch is reasonably easy to justify. He is being dragged off this male and could simply justify it as self defence under common law. He has a single punch and doesn't use any further force until the male gets up.
First punch to second male could also be justified as a pre-emptive strike either as self defence or under s3 criminal law act to prevent offences. The makes body language was threatening and it would be reasonable to apprehend immediate unlawful violence, especially as he was part of the melee at start of the video.
Second and third punches to the first male are a bit harder to justify, but within the the context of the situation and the fact the male is again getting up and confronting the bouncer would suggest that his intention is to be further violent. What helps the bouncer is that they are single strikes that are not continued until the other party reapproaches the bouncer. Some of the questions that would be asked of him would be whether he had a honestly held belief that he apprehended immediate unlawful violence. The fact that we know has been violent and continues to reapproach, I think that's reasonable. What other motivation would he have? Regardless of whether that's the male's actual intention, the bouncer would be reasonable in holding that belief.
The level of force he used in the subsequent punches seem a bit out of proportion but that's covered under s76 of the Criminal Justice and Immigration Act 2008. That basically states that the in the middle of a fight when you go tunnel visioned and miss so much information that a video or a bystander shows, you can't be expected to make perfectly proportionate responses and the actions should be judged on how you perceived them at the time. There's also clear case law that preemptive strikes are reasonable force if the person expects immediate unlawful violence. Many people will disagree but when the male gets up after the first punch, I think it's reasonable for him to be intending on a bit of perceived retribution.
If the bouncer explains his actions in these terms, I highly doubt you could ever get a charge.