英文処方箋は必要なのでは？

とあります。ともに、Class C Schedule 4pt2の薬です。

Travelling with medicine containing a controlled drug
You need to prove your medicine is prescribed to you if:
it contains a ‘controlled drug’
you have it on you when you’re entering or leaving the UK
You may also need to get a licence.

You could get a fine or go to prison if you travel with medicine that’s illegal in another country - check with the embassy of the country you’re going to before you travel.

Check if your medicine contains a controlled drug
Ask your doctor or pharmacist whether your medicine contains a controlled drug.

You can also check the drugs listed on the packaging of your medicine and search for them on the controlled drugs list. The list doesn’t give names of medicines, only drugs that are used in medicines.

If your medicine contains a drug listed as schedule 2, 3 or 4 on the controlled drugs list, you need to either:

prove it is prescribed to you
get a licence - if you’re travelling for at least 3 months or carrying enough to last you that long