There are no flights from the UK to Elba; the island's small airport is only served by flights from Switzerland. Several airlines, low-cost and regular, fly from the UK to Pisa, about 75 minutes' drive from Piombino for ferries to Elba (taking 40–60 minutes). You could also fly to Florence, about two hours from Piombino. If you want to take a car on the ferry, you need to book well in advance.
For train travel from the UK to Italy, see Man in Seat 61 (seat61.com).