Origin seems to recognise that I purchased NFS: Hot Pursuit on Steam, and is now giving me the option to download it. Is there a way, or is it possible, to install games in Origin that have been backed-up/downloaded via Steam?
I haven't installed Origin yet, but I would say it isn't even technically possible, short of reversing the game, Steamworks and the Origin interface.
Steam and Origin are different DRM systems with easily different interfaces between the game and the distribution system (i.e. ways for the program to ask "what's the user's avatar?", "does the user own this game?", "does the user have this DLC?", "please unlock this achievement", "please begin the purchase of this hat", etc.)
You can't just swap steamapp.dll for origindrm.dll (or whatever the relevant libraries are) and expect the program to work still.
The games could even be available in different versions between the two different platforms, so even just copy pasting the bulk of the game (levels, music, etc.) might be in vain.
In short, no, it's not possible.
The Origin Code Redemption FAQ lists a number of titles that can be re-redeemed through Origin. Need For Speed: Hot Pursuit is one of them, you'll be unsurprised to know. There's no information on what happens to the Steam version of the game, though. I actually asked that question here but strangely enough it was closed as a duplicate of this question :/
It is technically possible, but a solution does not appear to have been developed, and it is unlikely that it ever will.
There exist a multitude of ways in which conversion may be achieved, reverse-engineering the protection of Origin and Steam being one of the most complex. A delta patch would made by somebody in posession of both copies would do the trick and be fairly small, but the chances of somebody taking interest are slim.
It is highly probable, however, that the bulk of the files for both version will be the same, and there may be a way to make Origin recognize them and redownload the files that do not much, but as somebody how neither uses Origin nor has the game in question, I am unable to test this.