The CAA rules are here:
https://www.caa.co.uk/Passengers/Before-you-fly/Baggage/Items-that-are-allowed-in-baggage/
(The rest of the EU will have similar rules).
To save you some sums, a 5000mAh charger, delivering 5V is rated at 25Wh (and is thus well within the limits shown for most types of batteries).
So, as I read it, you can take 'power banks' in your cabin luggage.
Note there are additional (much stricter) rules on what you can take on flights to the UK originating in Turkey, Lebanon, Jordan, Egypt, Tunisia and Saudi Arabia.
A friend of mine flies regularly between Italy and the UK, with only hand luggage, and I know that she carries a power bank in her bag. It's never prevented any problems for her.