Will Android support WebSockets in an upcoming version?

Will Android support WebSockets in an upcoming version?

Probably, given Google's HTML5 emphasis.

If so, does anybody know which version it's scheduled to be supported in (in built-in Chrome Lite browser)?

Google does not publish that sort of detail in advance of releases. Hence, you'll know about it when it ships, not sooner.

Also, is it currently supported on any of the alternative browsers for Android like FireFox or Opera Mini?

Firefox Mobile's FAQ does not list it among the HTML5 features it presently supports. I have no idea about other browsers.

The iOS 4.2 beta currently has WebSockets support: http://twitpic.com/2yiygv

Come November when iOS 4.2 actually ships, if it still has WebSockets turned on (it has been in a previous beta and been turned off before shipping), then you can be sure that google won't be far behind.

Regardless, I predict that it will arrive with Gingerbread (the next one) since they are making such as big for other HTML5 features in that version: http://www.shoutpedia.com/what-is-next-to-froyo-android-2-3-might-be-released-by-fall-of-2010-3457/

Opera Mobile, Opera Mini and Firefox Mobile do not currently (Feb 2010) support WebSockets and won't do so until a change to the specification has been made. This is because a security issue was found in November 2010 in the underlying protocol: http://www.ietf.org/mail-archive/web/hybi/current/msg04744.html

I imagine WebKit support is also on hold until it's safe again, but it's not clear when that will be.