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.
politics.stackexchange.com/users/8554/James K
I would not overthink this.
I suspect it was nothing more than a desire to avoid getting involved in a war that had no upside for the UK.
However, Trump's willingness to abandon NATO means he is not going to find friends for causes that are not in the US's allies interests.
There is nothing to be gained by deposing Assad.
Look at what happened in Iraq, and Afghanistan.
Unless you are going to conquer the land for the spoils, war is a loss.
In the past the US's allies have backed the US interventions because they needed to know that the US would be there if they were attacked.
The fears of the cold war have been replaced by new fears now.
Those fears are not assuaged by the alliances that were formed to prevent another World War in the 1940s.
politics.stackexchange.com/users/169/SoylentGray
May's hesitance is as likely to come from domestic concerns as any geopolitical consideration.
There are two concerns in particular worthy of note: Theresa May is a weak leader of a minority government.
It is almost certain that the opposition leader Jeremy Corbyn will oppose strikes and likely invoke the whip to require Labour MPs to vote against it.
This means that May's ability to get parliamentary support for action in Syria is uncertain.
While targetted action is likely to attract the support of enough Labour rebels and others to gain a majority, any action that has the potential to lead to more serious military conflict will be more difficult.
She has more important battles to spend political capital on than Syria.
Since the fallout from the Iraq war, the UK public is not perceived as being supportive of further military action in the Middle East.
UK politicians are, accordingly, much less willing to support military action than they used to be.
politics.stackexchange.com/users/4713/Jack Aidley
The question assumes that NATO is relevant in the Syrian context.
NATO's role is explicitly limited to Europe and North America.
NATO has clarified that means Turkey can be a member, despite being only partially in Europe, and as a member the whole of Turkey is protected by NATO.
That means Syria becomes a NATO issue once Syrian troops cross the Turkish border, and Turkey asks for NATO support.
That simply hasn't happened.
Of course, NATO also has a logistical aspect - NATO militaries use common standards, which facilitate cooperation.
And such common standards still make sense in non-NATO operations.
So when the US and friends use NATO standards to coordinate actions above Syria, that still doesn't mean it's a NATO operation.
politics.stackexchange.com/users/10643/MSalters
The UK has a more precarious position with Russia than the US does.
Remember, Europe in general imports a LOT of Russian energy
Russia's gas exports to Europe rose 8.1 per cent last year to a record level of 193.9bn cubic metres (bcm), despite rising competition and concerns about the country’s dominance of supply.
State-run Gazprom, the world’s largest gas producer, has a monopoly over Russia’s network of pipelines to Europe and supplies close to 40 per cent of Europe’s gas.
But it has been forced to lower its prices in recent years to protect its market share in the face of moves by EU member states to buy more gas from the US, Qatar and other producers.
Getting into an armed conflict with one of your major energy suppliers is not on anybody's radar.
If you don't believe me, just ask Crimea .
politics.stackexchange.com/users/12027/Machavity
She did not refuse.
The U.K. joined the U.S. and France in military action against Syria on April 14.
British Prime Minister Theresa May described the strike as "limited and targeted."
She said she had authorized the British action after intelligence indicated Assad's government was responsible for the attack using chemical weapons in the Damascus suburb of Douma a week ago.
Source: Reuters
politics.stackexchange.com/users/4846/reirab
The UK is one of the main NATO members. It seems to me that Western countries almost always act in a block. Why is the UK stepping back from the line? Does she not want to burn jets in Russian fire?  In this question NATO is viewed not only as a military alliance, but mostly as a Western alliance of interests, where the UK has been almost everywhere side by side with the US.  Source:   The Times - May resists calls to join US action
What would have been the aim of Theresa May refusing to join the strike on Syria?
politics.stackexchange.com
<united-states><united-kingdom><donald-trump><syria>
politics.stackexchange.com/questions/30268
politics.stackexchange.com/users/20263/user2501323
This may have been nothing more than a desire to avoid getting involved in a war that had no upside for the UK.  May's hesitance is as likely to come from domestic concerns as any geopolitical consideration. The war wouldn’t have benefited the UK and it had to be careful in its dealings with Russia.
The war wouldn’t have benefited the UK and it had to be careful in its dealings with Russia.