Firstly: 

It is well-known that the basic principle with regard to a woman travelling without a mahram is that it is not allowed and is prohibited, because of the clear sound evidence that has been narrated concerning that. This has been discussed in a number of previous fatwas, such as no. 47029 and 145413. 

One of the conditions that must be met by the mahram is that he should be an adult, because the purpose behind this role is to protect the woman, and a minor cannot do that. This has been explained in fatwa no. 316 

Based on that, a child who has reached the age of nine years is not qualified to be a mahram for his mother when travelling. 

Secondly: 

The prohibition on a woman travelling without a mahram is a prohibition of the means (that may lead to negative results), not a prohibition of the aims and goals. In the case of this kind of prohibition, that which is prohibited may be made permissible in the case of necessity and overwhelming interests. It should not be applied so strictly that it puts severe pressure on a person and is detrimental to his interests. 

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: 

All of Islamic teaching is based on the idea that in the case of bad things that deserve to be prohibited, if they are countered by an overwhelming need, that which is ordinarily prohibited may become permissible.

End quote from Majmoo‘ al-Fataawa (29/49) 

He (may Allah have mercy on him) also said: 

That which comes under the heading of barring the means is only prohibited if there is no need for it. But when there is a need to serve an interest that cannot be met otherwise, then it is not prohibited.

End quote from Majmoo‘ al-Fataawa (23/214) 

Ibn al-Qayyim (may Allah have mercy on him) said: 

Whatever is prohibited so as to bar the means may be permitted in the case of an overwhelming interest, just as looking [i.e., a woman looking at a man] is permitted in the case of a marriage proposal, giving testimony, and medical consultation.

End quote from I‘laam al-Muwaqqi‘een (2/161) 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

Whatever is prohibited because it is a means (that may lead to evil), may be permitted in the case of necessity.

End quote from Manzoomah Usool al-Fiqh (p. 67) 

Based on this shar‘i principle, we have previously published on this website a number of fatwas in which it says that it is permissible – in some circumstances when there is a genuine need, or in order to achieve an important interest – for a woman to travel without a mahram. Please see fatwas no. 175260 and 181343. 

Conclusion:

There is nothing wrong with your wife travelling with her two small children, if you think it most likely that an important interest will be served by that, and the environment in your country to which your wife is going to travel is safer for the children and more likely to help them in terms of their religious commitment. 

However we should point out that it is important for her mahram to remain in the airport of departure until he is certain that she has boarded the plane and set off, then he should make sure that she will be welcomed by her mahram in the airport at her destination. 

Please see also the answer to question no. 122630 

And Allah knows best.