As of 1st of January 2021 (at 00h00 CET) UK nationals, who now enjoy travel rights as EEA nationals, will have the status of third-country nationals when the transition period following the UK´s departure from the EU ends on 31st of December 2020.
This means that UK nationals will be subject to the same entry requirements as other third country nationals, including in relation to COVID-19, and so Schengen related travel restrictions may apply.
NOTE however that UK nationals (and their family members) who reside in Iceland or have a residence permit or right of residence before 31st of December 2020 will maintain their residence rights after 1st of January 2021 in accordance with the Separation Agreement signed by the EEA EFTA states and the UK. Those UK nationals (and their family members) will therefore be able to travel to and from Iceland and travel restrictions due to Covid-19 will not apply to them. Until Icelandic authorities start issuing the new type of residence permit card for British citizens with the right to reside in Iceland prior to 1.1.2021, UK nationals benefiting from the Separation Agreement can proove their right of residence in Iceland with the Residence certificate (C-122) issued by Registers Iceland (Þjóðskrá).
For further information please go to the web of the Directorate of Immigration