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If you are a BN(O) status holder, you will be a qualifying commonwealth citizen with leave in the UK, and therefore will have full voter rights. However, if your dependents aren’t registered BN(O) status holders they may not qualify until they have reached citizenship, unless they hold another form of citizenship which would make them eligible.
You can find more information about how to register to vote on this website.
Your local MP is listed on the UK parliament website at: UK parliament website.
There has also been a recent Parliamentary Question on whether BN(O)s are able to vote in General Elections:
British Nationals Overseas (BN(O)s) who have arrived from Hong Kong are not required to become British Citizens in order to register to vote in UK general elections. Although BN(O)s are not classed as British citizens, the British Nationality Act 1981 determines that they have the status of Commonwealth citizens. This means that Hong Kong BN(O)s, with leave to enter or remain in the UK, are entitled to register to vote as an elector in UK parliamentary elections, providing they fulfil the age and residence requirements for such registration and are not subject to any other legal incapacity.