You are deeply confused, probably by the blogs of a conspiracy theorist (perhaps discussing the Sovereign Citizen Movement mentioned in the comments), whom it would be helpful for you to reference.
In fact, people with and without lawyers claim common law rights in the ordinary courts of the UK every day, in the lion's share of civil lawsuits. For example:
There is a common law right to sue for damages when someone breaches a contract by not paying a bill that they owe. A defendant, meanwhile, has a common law right to defend against such a suit on grounds, for example, that the debt has been paid or that the debt is not owed because there was no agreement to pay in the first place.
The substantive right of an owner of real property to evict a tenant who breaches a lease arises at common law, even though statutes spell out the process for enforcing that right. Furthermore, the way that ownership of real property is established (i.e. through a chain of title involving purchases by deeds) likewise arises at common law. The defendant meanwhile has a common law defense to a claim for rent for the remainder of the period in a lease after an eviction for failure of the landlord to mitigate damages if the landlord does not make a reasonable effort to find a new tenant.
The right to sue someone who negligently caused an accident that injured you is a common law right.