At the moment, you don't have to have a UK passport, it's up to you. You can travel with your original country's passport.

Things may change after the end of June 2021, when having status becomes mandatory for EEA nationals.

If a person is has a dual citizenship (UK and another EU member state) it is recommended that they apply for a British passport and they use the British passport to enter and exit the UK.

*Dual citizens are advised to travel with both passports - UK passport and EU member state passport - as many EU countries require their citizens to enter and exit their territory with the country passports, even when they have dual citizenship of another country.

Example - Dual citizens of Italy and UK, travelling from the UK to Italy via plane, will use the UK passport for the registration on the airline's form - for both flights to Italy and from Italy. At the passport control exiting the UK they will use their UK passport, on arrival in Italy and on departure from Italy they will use their Italian passport. On arrival in the UK they will use their UK passport.
You can still vote at General elections provided you register to vote or update the electoral register with your new citizenship if already registered.

Inside the UK, your Certificate of Naturalisation is proof of British citizenship, however, it is not a valid travel document.

Rules may be different for certain countries such as the Netherlands, where citizenship can be lost if living abroad and not renewing the passport. Therefore, it is important to check the rights and privileges of their own EU home country and not just take for granted what they read online, as within the EU each country still has its own laws.

See Dual citizenship.