In an extreme case, they could potentially blacklist your ID to prevent you from spreading "harmful" political opinions, cutting you off the web entirely.
It requires everyone to upload either ID and/or high quality photos & videos of themselves to a random company. Not just one company one whomever a website chooses for age verification, which can include doing it themselves. This creates multiple massive treasure troves of IDs that will attract hacking attempts (for example the Tea app breach). It creates opportunity for blackmail from this data (for example the Ashley Madison breach but much worse). For those age verification services that require a photo/video, that creates a resource for deep fakes. Plus any 15 year old boy worthy of their digital device will be able to get around age verification using fake id/photos or a VPN, whilst a less savvy adult trying to access information about quitting drinking or drug abuse will face a barrier.
And this is for ANY website that has a very broad range of content that the OSA mandates age verification for. It's easier for a website to err on the side of caution and just block the UK. That especially includes websites that have zero reporting back to Meta/Google/etc... for usual marketing profiling. If anything it pushes more people into the limited, monitored and advertising driven Meta/Google web.
Oh look, that foreign entity now has a profile of you, and what sites you've visited.
Fast forward a year, they get hacked (or maybe even just sold), a copy of your personally identifiable details are pinched along with your browsing history.
I really really hope you weren't being naive enough to go down the utterly stupid "nothing to hide" route with your line of thinking.
I'm far more bothered by this being yet another nail in the coffin of web forums. The issue is the burden the regulations place on any site which hosts user-generated content. For small low-risk sites the burden isn't all that great - OFCOM have some guidance on their website - but it's still enough to be offputting. A 100-user web forum isn't going to want to bother with it, so those run by UK nationals are just pulling the plug, while those run elsewhere are just blocking UK users because it's the easiest solution.
Which reminds me, I should go and disable comments on my blog...
Also, any hypothetical attempt to abuse the OSA to rein in political dissent would almost certainly be subject to legal challenges under the ECHR.