Are Great Britain and Italy parts of the same geological feature?
No

Great Britain doesn't necessarily have a unitary geological origin, but the mountains on it (such as there are mountains) are well worn left-overs from the Caledonian orogeny of 400 million+ years ago. These were thrust up before the tetrapods even took to land as part of the plate collision that formed the northern half of Pangaea. This is the same orogeny that made the Appalachians in the US and contributed to the mountains of Scandanavia, as well. 

The Appenines are a result of the much more recent Alpine orogeny. Caused by the collision of Africa into Eurasia over the last 60 million years or so, this orogeny built mountains from Spain and Morocco to the Zagros and Alborz in Iran.