This will be a meta-bug tracking all the changes, since this is a non-trivial project.

We need to:
1) First, deprecate the old wallpaper picker code after the new one is launched in stable.
2) Consider migrating to Polymer. In particular, the grid layout can be achieved in similar ways with the chrome://extensions page. The code point is at [1]
3) The other elements may reuse the shared components in [2]. However, the wallpaper picker introduces some new UI elements (e.g. a new button style). We should consider adding them to shared components as well.

[1] https://cs.chromium.org/chromium/src/chrome/browser/resources/md_extensions/item_list.js?dr=C&g=0

[2]
https://cs.chromium.org/chromium/src/ui/webui/resources/cr_elements/