As a family member of a UK national, if you are already registered as resident in Slovakia, or will do so by the end of the transition period (before 31 December 2020) and continue to reside here thereafter, your rights will be protected for as long as you remain resident. You will have to apply for a new Residence Card, even if you currently hold one, at the latest by 30 June 2021 (unless the transition period is extended).
In specific cases, your rights will be protected, and you will be able to join the UK national in Slovakia also after the end of the transition period (after 1 January 2021). This possibility applies only to some categories of persons.