What did the spy do with his suitcase radio?

No, Boyd's character and the events surrounding him are entirely an invention of the film. (I suppose the Germans might have sent a spy to investigate "Major Martin", but if so the British remained unaware of it.) So is Lucy, although a fake love letter was placed on the dead man's body. It was written by Montagu's secretary -- the Gloria Grahame character was another cinematic invention.

Operation Mincemeat has been the subject of debate and minsinformation since it became public knowledge when Montagu wrote his book in 1954. The manner of their finding a body to use, and who they used, as depicted in both the book and film, was false. Years later it was stated that the real corpse was that of a middle-aged homeless man who had died in a London hospital, whose body they simply used as he had no one to claim it, and this was the "corrected" tale accepted for two or three decades.

But after many more years the story now seems to be that the body was that of a British sailor who had been killed, along with all his shipmates, when his ship was mistakenly torpedoed by a British submarine off Scotland. Many bodies washed ahsore, and the incident was covered up by the government for decades, for obvious reasons. But this would also explain why it took so long for Montagu and his aide to drive to Southampton with the body. Even in 1943 the journey shouldn't have taken as long as it's depicted in the movie. (My wife and I actually drove a similar route just last month.) In point of fact, their trip took so long because they first secretly drove north to Scotland to retrieve a suitable corpse, then had to travel all the way back down Britain to rendevous with the submarine.

(Interestingly, despite all the fictionalizations in this film, some details remained true. The sub's commander was in fact named Jewell, for instance. He died about seven years ago at 90, having spent the last ten years of his life as a quadriplegic following a fall down a flight of stairs, a sad irony seeing as how he'd made it through the war unscathed.)

By the way, the real Ewan Montagu is in the movie. He's one of the Chiefs of Staff Webb and Naismith meet with to discuss the plan -- the one who says, when actor Alan Cuthbertson says he'd thought of four ways the plan could go wrong, "Four? I was going to say forty," and who near the end of the scene warns "Montagu" that if the Germans see through this, he's pinpointing Sicily as the actual target. As you can see, the real Montagu bore no resemblance to Clifton Webb whatsoever.