He or she cannot use the pre-nominal style of 'Sir' or 'Dame', but can use the post-nominal letters (after their names), subject to the prevailing conventions in his or her own country.
If such a person later acquires citizenship of a Commonwealth realm, then any honorary awards usually become substantive, and in the case of knights and dames they can begin to use the pre-nominal styles.
Citizens of British protectorates and mandated territories usually received honorary awards.
The effect is that, officially, the person is considered never to have been given the award, their name is erased from the order's register, and they are required to return all insignia.
This is not common and in practice only occurs where the recipient has engaged in serious detrimental activities, e.g. following a conviction for serious crimes, or for hostile acts against the United Kingdom.