In the event of the U.K. leaving the European Union without an agreement, the Polish Government has recently published a draft bill outlining the rights of U.K. citizens living in Poland before 29th of March 2019, when the U.K. is scheduled to leave the EU. The draft bill will only take effect once the U.K. leaves the EU and on condition that a no-deal Brexit scenario is given. However, the bill has not yet been approved by the Polish Government.
Among other provisions, the bill entails the right for all U.K. nationals who are residing in Poland under legal circumstances, (i.e. that the obligated registration application has been completed and a registration certificate has been received) to continue to have the right to work and follow economic activities in Poland also after 29th of March 2019.
On 30th of March 2019, a transition period of one year would kick in and be in full effect until 31st of March 2020. Within that period of time, all citizens of the U.K. holding a registration certificate as a national of the EU would be requested to complete an application which in turn would make them eligible for a temporary residence permit. This permit then is expected to be valid for three years. The application would be rejected if the registration certificate as a national of the EU is not included in the application and is therefore of great importance.
Citizens of the U.K. living in Poland who have never officially registered as an EU citizen would have to obtain the registration certificate before Brexit in order to keep the right to legally reside in Poland and also would need to make use of the transition period to submit the respective application.
An application for permanent residence as a third-country national would have to be submitted, regarding citizens of the U.K. holding permanent residence permits, after initially residing in the country for five years as a national of the EU. In this case the time frame of submission would be the same. It has to be done during the transition period or else the status of permanent residence would be taken.
The non-EEA/ non-EU visa issued to dependants of U.K. nationals would remain valid, provided that they are under the EEA/ EU national dependant family member route and valid beyond 30th of March 2019.

As previously mentioned, the draft bill is yet to be approved by the Polish Government, however, it becomes clear that all nationals of the U.K. residing in Poland are advised to complete an EU National registration and furthermore obtain a registration certificate prior to 29th of March 2019. If not, the risk is taken to lose the ability to apply for residency under the Brexit route and to be considered as a third-country national as soon as Brexit has officially kicked in.