According to Engadget it should be able to work, as the Oyster cards seem to operate by NFC. The article also mentions the ability to use other kinds of cards and other NFC tag based trinkets (NFC sticker, NFC wristband). However, these are all issued by financial institutions, so they would reasonably contain some kind of encryption that might be hard to integrate with off the shelf NFC tags.
If I were you I'd try scanning an Oyster card using an NFC reader application, like this one. If you find that the card returns a fixed value for each reading, then go ahead and try to program this value into a custom NFC tag, essentially copying the card. If this new "card" can be used with the official Oyster readers then you are all set. However, it probably won't work, but there is still hope (see next paragraph).
This video suggests that the Oyster cards use some kind of security chip (Mifare Classic Smartcard Chip). This is reinforced by Wikipedia too. It seems to be possible to buy this kind of chip from this website.