Running a startup in parallel with a job in the United Kingdom?

There is nothing wrong with running a side business while employed, however you'll have to check your contract of employment to see if there are any terms which prohibit you from doing other work, or which allocates the work to your employer (I see US people complain about that sometimes, but it is less common here in the UK). Such contract terms are not necessarily enforceable in court (due to restraint of trade, human rights, etc), but that doesn't mean you should do it if your employer will be unhappy, as that's a quick way to destroy your relationship with them.

You can tell HMRC that you are self-employed and then do an annual tax return to declare your additional income (if any). You might want to consider setting up a Ltd company though, as it provides protection to you in the case that something goes wrong (the "Limited" means that in general someone can sue the company but not you personally).

You can opt-out of the 48-hour rule and anything you do for reward is probably classed as "work", not a hobby, but it doesn't really matter anyway.

Good luck.

First, I would advise against not registering a company, it's too cheap nowadays not to protect yourself..
Second, think of it in simpler terms:
In order for your company to sue you, they have to first know about your project
If they find out about your project, beyond blabermouthing lol, means your project became successful.. At which point, one of 2 things happens, you are now able to hire a good lawyer and win the case.. Or you get a good lawyer and a crappy judge, and have to pay them 1% of your revenue for the rest of the company existence
One very important factor to consider is what the project entails. If it's not in direct competition/support of the company's product, I don't see any judge in their right mind swinging the case against you