Is it permissible for a husband to allow his recently converted Muslim wife to visit her sick non-Muslim grandmother in a non-Islamic country, given that there are security and personal concerns that prevent him from accompanying her?

It is not permissible for a wife to travel without a mahram or her husband. Seeing the grandmother is not a necessity that permits travel without a mahram. Modern means of communication suffice for that. If mischief is feared from the wife's travel, such as her family attempting to detain her and her daughter, then it is not permissible to allow her to travel even with a mahram. The husband should be wise in gently persuading his wife to forgo travel if it entails religious prohibitions. He is not permitted to hand over his daughter to his wife's family for fear of corrupting her faith.