I have been able to chat via my cell phone… it may be that chat is unavailable at the moment and therefore not showing up on your phone. When available, the chat icon shows on my phone in the same way it shows on my desktop.
Related comment: i have used Safari to chat on no problem and have had at least one chat unexpectedly terminated using Chrome due to the page reloading mid-chat (both on the same iPhone). Something to keep in mind…