The best way to proceed is to hire a structural engineer. Good luck.

Elmer's answer is the only answer.

Is the top of the exterior masonry wall at a height that beams can be supported with grout/shims or will pockets need to be created? It will be difficult to create pockets in an old rubble wall. I would think about running the beams the other way, which removed the requirement for a new bearing wall. A new bearing wall would require a proper concrete footing. I would plan on using metal deck over steel beams. The type of deck, the thickness and specification of the concrete, and the beam spacing/sizing would come from an engineer. Are the roads salted there in winter? There have been failures of structures like these after many years of salt corrosion.

First of all please see photo below, to get a better idea of the kind of barn. The new floor will be at the level where the wheel barrow is, next to the brown door. This is the upstairs area, there is another floor below, accessible from another side of the barn.

Removing the old beams will leave pockets, although those pockets will probably need to be enlarged / adapted for the new beams.

There's one cement coloured area in the photo which was from when I repaired a hole in the wall, using stones and cement.

I hope to find out all the structural advice from the relevant professional.

This location doesn't usually have much snow, normally less than around 10 days a year, so not a lot of salt on the roads. Also, I would be planning to have the floor well protected, maybe using some kind of suitable tiles on top of the embedded steel reinforced concrete and metal decking...all supported by new steel beams of course! Thanks