Suppose that radios (or magic with a similar effect) were available during the Medieval or Renaissance era. They're too expensive to give to every soldier or install in every town, but a knight might have a <1-watt backpack-sized radio, a better off noble might have a 10-watt unit pulled by a team of horses, and a large castle might have a 100-watt transmitter. How would this affect warfare and politics?
Oh the effect would be enormous.
As long as your ship has a radio, you can listen to it and find out when it's noon at a place. Then you figure out what time it is. Lo and behold, you have your longitude. It is difficult to understate how much impact this has on ease of navigation within radio range.
Prior to the telegraph, any criminal can move ahead of news of his crime, because all means of transportation were as slow as he was. Now you can send word ahead and have him arrested. Which leads to another factor.
Power that had to be delegated because the time lag would make it impossible can now be centralized. Burke, discussing the fractious American colonies, pointed out that the distance meant that months were lost in any clarification and necessarily entailed a certain looseness of control. Likewise, ambassadors don't get to make decisions that their governments might dislike once they get the news, because the ambassador can radio back and get instructions.
News from far-off markets arrives instantly. What's more, what could take months of back and forth between merchants in two towns because their letters took a week to travel now can be done in minutes.
Espionage becomes a much bigger problem. Prior to the telegraph, British newspapers freely covered when British troops embarked. With it -- during the Crimean War -- they realized that Russian agents would just telegraph the news. In medieval times, this would have been particularly useful when dealing with Viking raids.
The centralization issue hits again. A general can communicate with his superiors. Still more, an admiral. Much more centralization is possible, though it would have to be developed.
If radios as you describe them are manufactured in the renaissance, that implies a total revolution in materials science, metallurgy, physics, and manufacturing processes. There would be batteries, wires, transistors or vacuum tubes, speakers ...
This technology should be able to produce electric lights, electric motors (a generator in reverse), and much more. So it isn't really the renaissance as we knew it.
If radios are provided by time travelers, the interesting question will be how those travelers interact with the locals, and what else they change -- intentionally or unintentionally.
Finally there is the magic option, which might be taken halfway in stride if it can be protrayed as "miracles" or "relics."
On the tactical level, probably little difference in medieval times. Armies were not sufficiently articulated to make use of better communications. In the renaissance, there might be some battlefield utility, cutting through the "fog" of battle. But are the armies prepared to issue radios to scouts and not to knights?
On the strategic level, the answer by Mary lists some possibilities. But much would depend on who has the radio and who hasn't got it. Can a merchant go to the municipal radio station and send a message to another town, or is that a privilege for kings? Depends on how many radios there are.
Centralization and Bureaucracy
The whole point of having vassals is that the monarch cannot be everywhere in their realm to ensure that their policies and orders are properly carried out.
With (reasonably) cheap, instant communication, those layers of command and control are flattened. The discretion (freedom of action) of each layer is reduced.
The size of the monarch's staff grows enormously, as the volume of transmitted data is far greater than was previously transmitted in letters and dispatches. Courtiers become ministers, with offices and their own staffs (clerks and bureaucrats), plans, metrics, and coordination role.
Subordinate vassals whose data indicates that they are not meeting the monarch's intent can be identified and replaced...unless they are particularly persuasive.
It will be very, very difficult for vassals or courtiers to usurp the monarch or to foment a coup or uprising -- travel to secret meetings and payoffs will be hard to hide from the monarch's radio-equipped Secret Police.