We have mobile & desktop websites running on different sub-domains (m. and www.). We want our mobile content to be crawled and treated as the primary site. How should I handle Canonical and Alternate tags.?
Point your entire www site to the m site with canonical on each page.
It's also better of you can create a responsive site that works on desktop and mobile as most sites have moved away from the mobile sub domain protocol. You can do this by looking into meta viewport=1. If you can get your site to be responsive with viewport then you can 301 redirect your mobile subdomain to the www version.
I don't think what you're asking is what you really want to do.
If you are sure you only really want to have the mobile view used for all users, and get rid of the non-mobile template altogether (which is what you seem to be asking) then just use a 301 redirect assuming it's something simple like www.example.foo to m.example.foo
However I suspect you wanted to ask about how to add an alternate or mobile view in google search concole (or just Add a website property) without removing your previous desktop site.