No. Registration is not required for a will to be valid in England and Wales. Your will is legally binding whether or not it is registered.
That said, registering it with the National Will Register (run by Certainty at certainty.co.uk) makes it far easier for your executor to find after your death. It costs around £20 and only takes a few minutes. It does not change the legal status of the will; it simply creates a searchable record of where the original is kept.
It is not compulsory, but it is worth doing.
Ready to write your will? Takes 20 minutes. From £79.
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Complex estate? For blended families, overseas property, or business interests, please find a qualified solicitor.