The Space Trilogy/WMG

Mark Studdock is the Patient from The Screwtape Letters

• Nice try, but the Patient is a bachelor who lives with his mother and Studdock is married. Also, Screwtape takes place during the Second World War and That Hideous Strength takes place after the end of the War.
• That doesn't present an inconsistency:Patient Studdock got married and moved out after the war.
  • Uh, the Screwtape Letters ends with the patient dying, apparently from a bomb blast.

The Tramp whom Merlin encounters is Radagast

Out of the Silent Planet comes from a different continuity than the other two books

Out of the Silent Planet contains a "this is real, but all the names have been changed" disclaimer at the end. In Perelandra, Ransom's name truly being Ransom is a plot point.

• • Well, Ransom is not an uncommon name. The "Arthur" part could have been made up, as well as parts of his background and the area where he lived, while his surname is still Ransom.

That Hideous Strength was started before the events in it were finished

Thus, talk of a certain college in the present tense.

That Hideous Strength was written long after the events of the book.

Possibly After the End, depending on whether it was truly too late for Jane to fulfill a certain prophecy.

Jane continued to have clairvoyant visions after the events of the book.

And she passed them on to C. S. Lewis. (Literary Agent Hypothesis.) That's how Lewis the narrator can know what various villains were thinking just before they died: it was revealed to Jane in one of her dreams.