Can I legally offer my nephew an unpaid internship in the UK?

I think the nepotism aspect would actually help here, because it makes more credible the assertion that no valuable work is being done and rather that it is a genuine episode of "work experience" motivated purely by the desire to enrich your nephew's skills and experience.

There isn't a specific law against what you propose - akin perhaps to letting a young relative wander around the factory, fiddle with the tools and machines, perhaps make a few trinkets, and talk to the staff.

The core legal question is whether your relative is being implicitly employed or hired by the firm as a "worker". The main thing to avoid is any quid pro quo at all - any perception that your firm is expecting to gain value or retain output from your nephew's activity (including defraying any costs associated with providing the training or supervision, not just whether it is a net profitable episode overall).

Throwing away any code produced, once it has been discussed and exhausted any training purpose, is one way to be sure of this.

Subjectively though, I don't think this is a situation that would provoke the interest of any enforcement authorities.

Check out the the possibility of offering your nephew an "externship." Similar to what Weather Vane mentions in their comment, externships are periods of (often) unpaid employment where the purpose is on education and exposure, rather than actually producing work.

In the states at least it is a fairly common thing especially when it involves nepotism, since this is a way to do someone you know a favor while not risking harm to the company or replacing a more deserving intern candidate.