I have read your post several times and I think, that the question that you are really asking is, can a clot form in the damaged area (forearm) and travel to the lungs ( Pulmonary Embolism)
The answer is very rarely a deep vein in the arm may become blocked by a blood clot, which may then break away to lodge in the lungs. This is an extremely uncommon occurrence.
\\\.My son works in a warehouse and works very hard, so maybe that's the cause. \\\
Yes, I agree that his work environment may have causes his problem and i agree with the diagnosis of the Casualty Dr, although I would question the prescription of antibiotics, but as he has improved then the method justified the means.
The danger injury would be to a deep vein and there is no suggestion of this in the above account, merely superficial vein problem.
As the redness and swelling has improved, then i would suggest that he is on his way to an uneventful recovery.