One big theme in Billy Elliot is
overcoming adversity. We see Billy rising beyond the confines of his class and family status through pursuing his dream of becoming a ballet dancer. You witness the sacrifices made by those around him (like his father's willingness to sacrifice his pride and unity to those around him by returning to work and becoming a scab in order to pay for Billy's school) and how those losses contributed to Billy's success. We as an audience also realise that not all are able to overcome adversity as Billy's fulfilled dream is juxtaposed by the end of the miner's strike and them returning to poor wages and working conditions. However, it also shows how hard work and passion will often pay off in the pursue of a dream.
You also have a few similar themes like
challenging stereotypes, class, and gender. These apply heavily to the context as we look at the stiff views of gender (and how Michael's character contrasts this greatly, as well as Billy's love of ballet, as a traditionally female art form) and the tense class warfare in the film especially between the working class (Billy's family) and middle class (Mrs Wilkinson's family)
The film also addresses
loss, rebellion, and pursing a dreamHope this helps!