The loyalty badges will be present in the existing badges tag. So you’ll see things like badges=subscriber/6 or badges=subscriber/12. The images can be retrieved the same way all other badge images are retrieved, via https://badges.twitch.tv/v1/badges/global/display and https://badges.twitch.tv/v1/badges/channels/:channel_id/display
See [beta] Badge API
There are no plans to include the actual number of months subscribed in the subscriber tag, at least not at launch. So it will remain a 0 or 1 tag.