He was born in Lewisham , a suburb of London. His father was a bank clerk. He first went to a school in Lewisham , but was so good at maths that he was given a free place at the famous Eton college, a boarding school where only the wealthy sent their sons.
There he was also good at maths and won prizes in the subject. He got a place at Cambidge University and again was better than his fellow students leaving with the best degree, a first class degree. 
When America began to secretly build the atomic bomb these two men and a few other Britons went to America to help them achieve their goal. Tony Skyrme did important work towards the two atomic bombs that ended the war.
He then worked back in Britain on studying atoms . He made up a way of understanding how atoms work by inventing an atom sized thing called a skyrmion which is not an atom but helps to understand the way atoms work. It is still used today to understand atoms.