"what happens to the speeds Vx and Vy with increasing altitude for a given TAS?"
I am with Keith and Alex, curious as this question makes no sense at all. Vx and Vy are both CASs. Therefore they are not dependent upon TAS. Also Vy reduces and Vx increases with increasing density altitude, until the absolute ceiling where they are identical.
Was the question about rates and angles of climb, in which case both decrease?
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Not concerning the main topic, but Spitfire something you said would have my FI(R) ground instructor spinning in his grave were he not still alive, as it was something he is always trying to irradicate as a misconception. You should not have been taught this at any stage, let alone ATPL!
"... stalled hence no lift and no induced drag ..."
A stalled aircraft is producing lift, and is producing induced drag. Draw a lift coefficient curve. The highest point is the stalling point. What happens after this? A stall is not a sudden, immediate loss of all lift. What happens is a reducing lift coefficient with AoA and a rapid increase in drag, that is a stall!