Could the Royal Navy name a ship HMS Anzac?

Would Australian restrictions on the name Anzac apply to a hypothetical HMS Anzac?

No; Australian law would not bind the Royal Navy, although an Australian HMAS Anzac is the more likely real-world case.

Australian legal protection of the name Anzac could not ordinarily prevent the Royal Navy from commissioning an HMS Anzac because one country cannot legislate for another sovereign state. A British use of the name would therefore fall outside Australian law. In practice, a future ship bearing the name is more likely to be Australian and commissioned as HMAS Anzac. The legal possibility and the likely real-world naming choice are separate questions.