The IRCJSA fare is, according to ba.com, refundable and changeable after departure without penalty.

What to be aware of:
- Changing the first flight before departure will result in the whole trip being repriced (and this fare may or not be available when you do this)
- Changing the subsequent legs before departure will result in the whole trip being repriced at the ticket prices in force on the date you bought the ticket (presumably no change if the routing stays the same, no stopovers added, and you book into the same booking classes - J/I/I/J).
- Adding a stopover (only 1 permitted) is going to add UK departure tax.
- Making any (free) changes after departure is going to require I class on the LHR-BKK-LHR legs.

My advice for the starting leg:
Fly Italy-LHR the day before, if possible, but with less than 24 hours. This will work on the BA547 from FCO ar 2025, arr LHR 2205.
Fly outward LHR-Italy either on the same plane (BA 546 leaving LHR at 1610, arr FCO 1935) or in the morning with the day in Italy, to minimise the risk of missing the turnaround.