Swords are off-limits:

https://www.legislation.gov.uk/ukpga/Eliz2/1-2/14/section/1

Prevention of Crime Act 1953

Any person who without lawful authority or reasonable excuse, the proof whereof shall lie on him, has with him in any public place any offensive weapon shall be guilty of an offence

That being said:

Shields also

I'm not aware of anything restricting armor use in public in the UK.

kagis

https://www.uk.safeguardclothing.com/blogs/articles/body-armour-uk-law

UNITED KINGDOM

In the United Kingdom, there are currently no legal restrictions on the purchase and ownership of body armour.

There is a law against wearing it in Parliament.

https://en.wikipedia.org/wiki/Statute_forbidding_Bearing_of_Armour

The Statute forbidding Bearing of Armour (7 Edw. 2. St. 1) or Coming Armed to Parliament Act 1313 (originally titled Statuto sup' Arportam'to Armor or Statutum de Defensione portandi Arma) was enacted in 1313 during the reign of Edward II of England. It decrees "that in all Parliaments, Treatises and other Assemblies, which should be made in the Realm of England for ever, that every Man shall come without all Force and Armour". The statute, which was written in the Anglo-Norman language, goes on to assert the royal power to "defend Force of Armour, and all other Force against our Peace, at all Times when it shall please Us, and to punish them which shall do contrary." It declares that "Prelates, Earls, Barons, and the Commonalty of our Realm... are bound to aid Us as their Sovereign Lord at all Seasons, when need shall be."[1]

The law is still in force today, though the Crown Prosecution Service has said that it is unaware of anyone being prosecuted under this or other archaic statutes in recent times.[4] According to a CPS spokeswoman, "If anyone was caught in the Houses of Parliament wearing armour it would first be a matter for the police."[4]