Q: does the iPad mini GSM have LTE

Yes. See:

http://support.apple.com/kb/SP661

Scroll down to "Wireless and Cellular"

Are you in the US or another country? I believe LTE is not available outside of North America. See:

http://appleinsider.com/articles/12/03/08/new_ipad_4g_lte_incompatible_with_netw orks_outside_north_america

Yes, but the iPad will only connect via 3G

You may want to check out the link below as well. The first link was from last year. The following is recent so I have no idea what the situation is over in the UK:

http://crave.cnet.co.uk/mobiles/ipad-air-and-ipad-mini-go-4g-on-ee-vodafone-5001 2578/

http://www.apple.com/ipad/LTE/

Compatibility with LTE worldwide by model