Physical activity and healthy diets in early childhood are associated with better cognitive outcomes in young children” (Tandon et al., 2016).

"The early childhood years are critical for both obesity prevention and neurocognitive development" (Tandon et al., 2016).

Physical activity, both aerobic and vigorous, were positively associated with cognitive functioning in children and the evidence also supports both short-term and long-term benefits (Tandon et al., 2016).  

There is a positive relationship between a healthy diet in early childhood, which also has cognitive benefits (Tandon et al., 2016).

"Both stress and neuroticism were associated with diet quality, but diet quality in turn did not predict the development of depression/anxiety... " (Shweren et al., 2021).
In other words, your diet is not necessarily the cause of your depression/anxiety. 

"Dietary changes are unlikely to, in the long-term, prevent the onset or recurrence of depression/anxiety in the general population" (Shweren et al., 2021). 

So, diet does not prevent the development of depression and anxiety, but there are still associations between diet and the mental health of individuals. 

We know that your diet (healthy or not) does not prevent the onset, but it is possible that a healthier diet may be able to help alleviate present feelings of stress/anxiety. 
If you are feeling stressed, anxious, or depressed, take a look at your diet. Food is our body's fuel and a healthy diet may be the best medicine for poor mental health.