Is it correct to say that true compassion (not trying to be) only manifest when there is wisdom? That is to say that wisdom must come first before compassion.
Correct. True compassion comes from being able to relate, to see things from the perspective of the object of compassion. While this may partially exist before wisdom, only complete realization of Emptiness gives unlimited insight into other perspectives. Normally, this ability is hindered by ego's attachments, biases, fixed views etc. The enlightened mind is completely free from these and can truly see from all perspectives, without distortion. This "knowledge of all modes" is what gives rise to true compassion.

Although, of course partial wisdom and partial compassion can develop together in lock-step.