Three do 4G free with all contracts, so you could get unlimited 4G LTE with tethering on The One Plan for £15 a month. Stick a phone that does tethering nearby (an iPhone 5/5c/5s, or whatever silly Android phone is doing LTE this week), enable WiFi tethering, and Robert’s your mother’s brother.
Granted, LTE in something other than a phone is a bit expensive if you don’t like data caps, but 3 are still pretty cheap. They hang off EE cells where they don’t have coverage, so YMMV dependent on what frequencies your device supports. Their coverage is increasing slowly but surely, and Birmingham was one of their launch sites last year.
Depends on what you define as “home broadband”. Regular ADSL2+, sure; VDSL2 (“Infinity”, et al) and FTTP, almost.