Is it a dry hacking cough, or one producing phlegm? That's the tracheal cartiliage area (below the larynx) that consists of rings of cartilage to protect the esophagus. How's your voice doing?
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Do you have patches on your throat like this?
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The tonsils, are they inflamed and covered with patches that are milky or yellowish?

If so consider Strep throat might be the cause.Warm saline can help remove them but don't swallow it. Spit it out.

Does is seem like there are swollen circular moveable places in them on the sides of the neck?

If so, you might need antibiotics. You're in England, maybe the chemist (pharmacist) could prescribe something.