Given that the Ghassanids and the Lakhmids served as buffers to prevent raids from the Arab tribes, and engaged in regular diplomatic relations with the ERE and the Persians, both empires must have been aware of the events happening to the south of those kingdoms. If they were aware, before the fall of Ghassan, did they percieve the conquests of Mohammed as a threat? If so what actions were taken by the empires to counter the threat?
It is important to keep in mind the Byzantines and Persians had been fighting each other for some time before.
We have contemporary reactions from Christian authors. Heterodox generally had a better opinion of Mohammed than the Chalcedonians.
The Byzantines before Mohammed saw Arabia as an area that would eventually become christianized like all the other areas, as part of their idea of history, with Christianity taking over the world gradually. The Islamic conquests brought this idea to an end.
To answer your question, whether things like the conquest of Mecca were observed, most of the sources we have come from after Islamic conquests had reached Byzantine and Persian territory.
Sorry if I didn't answer, I'll see if I can include sources tomorrow.
EDIT: (more)
We have the Arabs as a unified force for the first time, unprecedented in history. Reactions afterwards acknowledge the ability of Mohammed to unify the tribes, and saw them as a formidable enemy.
There is no evidence the two great powers cared about what was happening in the Hijaz at this time. Rome and Persia were fighting their last great war against each other, a war that lasted from 602 to 628, precisely when Muhammad's confederation in Arabia was formed. The Romans certainly had very little ability to intervene even if they cared that much - all their eastern territories had been occupied by the Persians in the 610s, so it was exceedingly difficult for them to influence events in the Arabian Peninsula. Both the Ghassanids and the Lakhmids had fallen from favour by this point as well, the Ghassanids in the late sixth century under the emperor Maurice and Lakhmids around 602 under Khusro II; the two empires still had many Arab clients on their borders and Persia even had formal or informal control of Yemen, so to a certain extent they would have known about the events in Western Arabia, but their strategic priority throughout this period was definitely their war against each other. The war was a truly devastating one for both empires, so we should not underestimate how important it was in shaping the rise of Islam. I am sure there were still ways for the Romans and the Persians to influence things, but we have no evidence for this, presumably because it wasn't important enough to be mentioned in the contemporary sources.
Our sources are also deeply problematic. The early sources for Muhammad and the rise of Islam consists basically of the Qur'an, the Constitution of Medina and various Christian sources from when Arabs attacked Rome and Persia. The Qur'an tells us that Muhammad was preaching a new message and that he was attacked by some people, that there was a migration of some sort to Medina, and that there were various battles - to be honest it doesn't really give us any specific details about the events historians are interested in. The Constitution survives from much later but its archaic language suggests that it was from much earlier and that it is an authentic document about the community at Medina from Muhammad's life-time, and you can read it for yourself here. As for the Christian sources, I quoted pretty much all the important ones here. Again, it doesn't tell us much, since contemporaries were not interested in how the Arabs were united or the reactions of the great powers. I did manage to write a fairly big post on Christian reactions to the rise of Islam, but it was cobbled together from all sorts of sources from across the Christian world, which is I think the only way to answer this sort of question - take a look here if you are interested! To get a general sense of the Arab Conquest and why it happened, take a look here, and a more specific comment on the conquest of North Africa here. Do always remind yourself that there is very little certainty about this period and to take my comments with a pinch of salt - new work is always being produced, especially in using the archaeological evidence and re-examining old assumptions, and our understanding of the seventh century is constantly changing.