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This is applicable to the EU:
Passports for British citizens are valid for travel to any EU country up to and including the passport expiry date.
(Source)
You can travel on the expiry date depending on the country, some require a certain amount of validity on the passport. In the EU there is no such requirement:
All 28 countries in the European Union require that your passport is valid for the duration of your stay.
(Source)
The same applies with ID cards:
We can only accept a EU Identity Card that is valid and within its expiry date. This would be the same for any identification documents. All would need to be within their expiry dates.
(Source)