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You can't get your copper phone line itself upgraded.

Openreach are offering a FTTPoD service where you can pay the install costs to have the copper phone line supplemented with a pure fibre connection to the home. The installation costs of this are estimated around £500-1500 but I'd expect yours to be at the high end if not higher. Groups of neighbours could band together and share the costs though, which would drastically reduce the per-customer cost and only slightly increase the total.

This is a long term investment though, that can dramatically increase home value, as it offers 1Gbps speeds now and up to 10Gbps in future.

3G dongles are OK for browsing and what not but not gaming. Same applies for 4G although it's slightly better.

There are no unlimited 3G or 4G dongles, although T-Mobile offer a semi-unlimited dongle plan.

3 offer a 3G mobile phone plan where you can use tethering and unlimited data, and technically although not morally this could be used in a dongle as well. This will include a free upgrade to 4G once it's available. However 3 have by far the least 4G coverage right now (actually, none) but should be catching up over the next year or so. That's the closest you'll ever get to an "unlimited" 4G service that can be used in a dongle.

Satellite is completely unusable for gaming and pretty poor for video services too due to paltry data limits. 4G even without unlimited will surpass it in every way.