If I only have Polish passport and Polish ID and live in UK but both my Polish documents expire.
What can I do to renew my any Polish document if I need to travel again? I do not have any British identification at all.
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As a national of any country you can go to your nearest consulate or embassy and get your expired passport renewed, for a fixed fee, without providing any other valid identification.
You do not need any other valid identification, as the expired passport already contains your identification numbers and will be reverified when a new one is created. You can pay extra to expedited the process or apply for an emergency passport which may be issued the same day or within 48 hours.
If you have lost all your ID's and passports, you must go to the nearest embassy, where the process is completely different and slightly unpleasant.
UK and other authorities will need you to procure another ID for any official purposes, including traveling. If you still have recently expired documents, renewing them is much easier and it might even be possible to travel in some cases.
If you don't, your country's consular network should help you (same thing if you have only one ID and you lose it). Depending on how your country works, they can do that by asking personal details and checking with the relevant local authorities back home (e.g. look up your biometric details in a national file, find out if there is someone matching your details in the registry of births, getting others to vouch for you, etc.) Registering with your embassy beforehand, if your country offers it, might also simplify things down the line.