There does not appear to be a transfer feature to convert an Xbox 360 Zoo Tycoon save file for use in the Xbox One version of Zoo Tycoon.
Examples of Save File Transfer
We know that such a feature can be implemented, as it has for such games as Destiny, Borderlands and Grand Theft Auto V. However, there are a few observant similarities in the games that do offer the feature.
- The feature is often advertised for player characters: In situations where the player can transfer a save, they are not necessarily transferring all of their safe progress. Whenever I see the feature advertised, it is directly advertised to players wishing to continue completing the main story line on the new console; it is advertised for players wishing to continue playing their character on the new console.
- The feature is often advertised where the upgrade only improves performance: In most cases, upgrading the game from an Xbox 360 version to an Xbox One version improves graphics and performance, but not much else. This might mean things are going on on the screen, but they are still the same things that occur in lower frequency, on the previous generation of console. Zoo Tycoon actually added a considerable amount of content, in the Xbox One version. There are 40+ new animals, and an entire online function. Some would consider the Xbox 360 Zoo Tycoon and the Xbox One Zoo Tycoon completely different games, in the same way that they are different from the PC version of Zoo Tycoon.
- The feature is often advertised to continue multiplayer features: In most cases, safe file transfer is advertised in order for players to continue playing the same character, in a multiplayer environment. Zoo Tycoon, however, does not have a multiplayer feature. Or rather, it does, but only in the Xbox One version; you would not have a multiplayer save file if you were playing the game on Xbox 360.
4 . Save transferring is run through the developer: in almost every situation, you have to complete the safe transfer process through the developer, externally. It is a process designed by the developers, and not really a feature that can be "turned on" for other Xbox 360 games, like it can be for backwards compatibility. This does nothing to explain why the feature is absent, but it means that if you think the feature should be implemented, you should probably direct your suggestions to the developer, Frontier developments.
The Future Prospects of Backwards Compatability
At the present time, the Xbox 360 version of Zoo Tycoon is not backwards compatible. If at any point it becomes compatible with the Xbox One, you can at least transfer your original Xbox 360 save files to your console, through the cloud. This save will not work on the Xbox One version, but might allow you to continue playing your older save, should an Xbox One be more accessible than the discontinued Xbox 360.
Achievements
Achievements have always been tied to the version of the game, while progress is tied to the save file. There are separate achievement lists for the Xbox 360 version of Zoo Tycoon and the Xbox One version of Zoo Tycoon, and they are subtly different. Even in the case of transferring saves, you will not be given the Xbox One version of the achievements you unlocked on Xbox 360.
You would still have your Xbox 360 achievements, but they would be listed under a separate Zoo Tycoon icon. You can also unlock the achievements, again, on the Xbox One version and both versions of the game will contribute to your overall gamerscore.
This entirely depends on the game.
For most Xbox Live purchases, the answer is no. Since the Xbox 360 and Xbox One have a completely different architecture, binaries are not compatible between them. For a game to be released on the Xbox 360 to be released on the Xbox One, it must be specifically released for the Xbox One. Due to this, many games available on the Xbox 360 Marketplace simply are not available for the Xbox One.
Some games offer a free upgrade. For example, Call of Duty: Advanced Warfare allows people who purchased the Xbox 360 version of the game to upgrade to the Xbox One version for free. The same goes for PS3 players, they will get the PS4 upgrade free of charge.
Other games offer an upgrade for a reduced price. An example would be Minecraft:
Owners of Minecraft: Xbox 360 Edition can upgrade to Minecraft: Xbox One Edition with a substantial discount. All downloadable content previously purchased for the Xbox 360 Edition is available free for redownload on Xbox One.
Finally, some games you simply must purchase again. Plenty of games have been developed towards both platforms, and have released for both the Xbox 360 and Xbox One. These games you would have to purchase again if you wanted to "upgrade" to the Xbox One.
The games you mentioned (Walking Dead 1 & 2 and Wolf Among Us) fall under this category. While the games have been re-released for the Xbox One, it is a full purchase. There is currently no free or discounted upgrade program for these games.
Best Answer
The Xbox One can't play Xbox 360 games at all due to using a different system architecture.
Both Microsoft and Sony made conscious decisions to not try to support the previous generation via emulation this time around.
Edit: Electronic Arts has a service named EA Access that can be used to play certain EA titles on the Xbox One including some older titles. It requires a monthly subscription fee, though.
Edit 2: This answer is very out of date. See TZHX's answer for the current state of things.