My parents live in Derbyshire, can get cold too. Cordylines do well in their garden. Bamboo isnt realy a tropical plant but creates a tropical look and can handle cold weather. Ferns another tropical looking plant that can handle colder weather & of course Hostas.
Bananas Musa species Musa basjoo (Japanese banana) which is the hardiest and can withstand down to around -8°C
Fatsia japonica (False Castor Oil Plant) Looks tropical but actually very hardy - -18°C
Trachycarpus fortunei (Chusan Palm or Chinese Windmill Palm) Very hardy - can even withstand down to around -15°C -
Chamerops humilis (Mediterranean Dwarf Fan Palm) Can withstand down to around -10°C (14°F).
Phormium tenax (New Zealand Flax) Hardy down to around -5°C
Gunnera manicata Very hardy - down to around -20°C (-10°F). Likes moist spots and needs a lot of space -
Gunnera tinctoria which is hardy down to around -15°C (5°F) and a little smaller than the Gunner manicata
You may need to have them sheltered & use some winter cover
Some people may dissagree but I think it depends on the temperatures in your area.