If someone in the 14-15th century world had the knowledge required to build an engine for use in locomotives and ships, what kind of engine could be made using technology that already existed back then?
But the first recorded steam engine was as early as the 1st century Egypt so the principles were known 1500–1600 years before your time-frame which means it's far from a great stretch for you to say development of steam engines that entered common usage was a little earlier in your world.
The propeller didn't come along for a while longer than that of course but paddle steamers are simply a reverse water wheel (which the Romans had) and Archimedes' screw also existed then, so that gives you two potential methods of transferring power from the engine to propel your ships.
Paddle steamers are your most likely ships, especially as the Romans already had ox powered paddle boats (there's a reference to them in the link).
Though this horizontal design is a touch more elegant to my eye.
They'd probably be wood burning rather than coal.
in the 1300s the only sources of power that could be mobile were people and animals. Using animals on ships was almost never done because it was wildly impractical to have treadmills taking up needed cargo space and food to keep them well enough fed to be able to work.
The metallurgy needed for harnessing steam was not available in this time period, now were the machining tools needed. Lathes were not invented until the late 1700s and even an expert blacksmith would not be able to make a piston using hand tools
Steam engines can be made of brass or bronze, both horribly expensive in the 13-1400s, but only in small scale, like in model engines. These materials cannot handle the sort of pressure needed to drive pistons well enough to overcome the friction of a propeller in water, and they cannot be made powerful enough for a land vehicle able to transport any great weight. Turbines .. well no .. just no.
The Conneticut Yankee in King Arthurs court took years to build the infrastructure that allowed him to build 19th century weapons and electrical equipment .. but building the entire tool chain took him decades.
so if someone with all of the needed branches of knowledge, and a wide ranging practical experience of manufacturing, does wind up in the 1350 wanting to build a ship bigger than the Santa Maria (19 m x 9 m ) he would have to be an expert negotiator and manipulator to make it happen.