I think that should be fine.
I would phrase it as LHR-JFK JFK-FCO though. (Rather than looking like you want to tag on a free extra flight). The distance between LHR and FCO is less than the distance of the long leg and it is in the same zone (Europe)
I did a MAN-SCL GIG-MAN on this voucher and the distance between SCL and GIG is twice as far.
All the flights need to be on BA so you should end up going through LHR where I presume you would like a layover?

The rules state
11. Cabin Upgrade Vouchers can only be used for British Airways flight-only itineraries with departures from the United Kingdom, and are available for British Airways mainline and British Airways CityFlyer and EuroFlyer franchise operated flights issued with a British Airways flight number only. Partner and code-share flights are excluded and will not be eligible for the Cabin Upgrade Voucher, including SUN-AIR.

You *are* departing from the UK.
It doesn't say you have to return.

For the avoidance of doubt, you cannot add your own extra flights on to the reward booked using the voucher even if they are Avios flights (I know, I tried). So you would need to book the return as a single journey.
I suspect the intent is to only allow you to come back to the UK, but it's not there in the T&Cs.