While deliberately infecting someone with HIV can constitute 'GBH'
http://www.cps.gov.uk/legal/h_to_k/intentional_or_reckless_sexual_transmission_of_infection_guidance/
there's no specific offence of either 'threatening to infect' or 'threatening GBH'.
A threat to cause violence (including 'GBH', whether by infection or otherwise) would most usually be dealt with under Section 4 (or, for persistent threats, Section 4a) of the Criminal Justice Act 1967, relating to 'public order'. However any threat made within a 'dwelling place' is exempt from the provisions of that legislation.
Further, if the threat was made electronically (by phone, text or email), separate legislation (relating to 'malicious communications') might be used to secure a conviction before the courts. (Perhaps that's why the contents of your phone, computer, etc are considered potentially relevant by the police?)