As a general rule, you do not have to do very much to secure your residence status:

UK nationals

If you are a UK national who was living in Germany on 31 December 2020 and are still living in Germany, you must report your residence to the foreigners authority responsible for your place of residence in order to be able to obtain the new residence document. In many places you can do this online. You do not need to make an application. Registration with the residents’ registration office is not sufficient! We recommend that you report your residence immediately.

Note:
Residence permits issued to you as a precaution in anticipation of a "hard Brexit" and residence documents pre-dating the accession of the United Kingdom to the then European Economic Community do not meet the requirement to report to the foreigners authority. However, if you have already reported your residence to the foreigners authority in anticipation of Brexit, you are not required to do so again.

Family members (including UK nationals who do not themselves fall under the Withdrawal Agreement) who already have a German residence document

If you already have a residence card or permanent residence card, you are not required to do anything on your own initiative. The foreigners authority will contact you. Your document will be exchanged for another document.

Family members (including UK nationals who do not themselves fall under the Withdrawal Agreement) who do not yet have a German residence document

If you have relocated to Germany from the United Kingdom after 1 January 2021 or plan to do so and if you are eligible to do so under the Withdrawal Agreement, you will generally need a visa before moving to Germany if you are a third-country national.

Exceptions: That does not apply to citizens of Australia, Canada, Israel, Japan, the Republic of Korea, New Zealand and the United States of America and for UK nationals (“British Citizens” and certain holders of other categories of UK citizenship from the Channel Islands, the Isle of Man or Gibraltar). These nationals must apply for a residence document from the foreigners authority within three months of entry. However, they may also voluntarily apply for a visa prior to entry, particularly to be able to demonstrate their right to pursue a paid occupation immediately after entry.

UK nationals working, but not living, in Germany on 31 December 2020

As a frontier worker, you may have rights under the Withdrawal Agreement and must apply for a special document in the form of a card from the foreigners authority responsible for your place of work in order to continue to be able to work in Germany (more).

The following rules apply for everyone:

The foreigners authorities provide forms or online platforms for the relevant reports. For persons exercising their right to subsequent immigration in 2021, the deadline for reporting their residence is three months upon entry (and ends in no case before 30 June 2021).

You can also apply for a certificate attesting to an existing right of permanent residence. To do this, you must provide clear evidence to the foreigners authorityresponsible for you that you have already lived in Germany for five years.