First let me mention your question "Does she not want to burn jets in Russian Fire?"
No she does not want to British Jets to be shot down by Russians. It is a silly question. No sane leader engages in an action with the intent of having her own troops killed.
Next you talk about "Nato" action. While there is a good deal of unity in Nato and other Western countries, the action in Syria is a US military action, not a Nato joint command. The UK has no treaty obligation to act.
Now, In 2013 the UK parliament had a vote on joining US action in Syria. By a narrow margin the government was defeated. It was the opinion of Members of the House that military action in Syria would lead to more unnecessary bloodshed. While nobody in the government had any support for Assad who seemed intent on bombing and gassing the Syrians into submission, there was a lack of a credible exit plan. There was a plan for war, but no clear plan for peace. The UK government has gone on to get approval for airstrikes in Daesh controlled Syria, but not for a general war with the Syrian Government.
The individual countries in the Nato, and the Western Alliance more generally can choose if a particular action is in their interest, unless one of the members of Nato is attacked and invokes "article 5". The French and German militaries did not join the UK and US in Iraq. The UK did not join France and the US in airstrikes on Syria in 2013.
May has the authority to commit UK forces to action. But she would seek Parliamentary approval to engage in war in Syria. The potential benefits of perhaps removing a very dangerous leader would have to be balanced against the risks to British troops, and the risk of insurgency and a spread of terrorism.
At the end of the day, Theresa May is not joining an attack on the Syrian Government, because she does not think it to be in the UK's best interest.