You won't make it. You have only allowed yourself 80 minutes to get to your train from the plane. What happens if your plane is delayed at either end (not an infrequent occurrence at either FCO or LHR)?

If you take the Piccadilly line, it would take about 60 min to get to Euston. Even with HEX, it would take about 30 min to get to Euston if the HEX leaves right away (which it may not). You could potentially be waiting 15 min for the train to leave.
This is pure train travel time and does not take into account off-loading from the plane and retrieving your luggage. This alone could take 30 minutes.

GBR is not part of the Schengen zone so I think you may have to go thru immigration coming from Italy. So add time for that as well.

Better to play it safe and save stress by taking the later train.