Via Francigena runs from Switzerland down through Italy to Rome, and your nearest airport and route to the start depends on which stage, or stages, you're walking.
If you choose the Full Italian Via Francigena tour (Bourg St Pierre to Rome): The best airport for Bourg St Pierre is Geneva, served by many airlines from the UK and further afield. From Geneva Airport, take a direct train to Martigny, then a bus from the train station in Martigny to Bourg St Pierre, with schedules and fares at the Trenitalia website. For your return, it's best to fly out of Rome, which has two main airports, Fiumicino and Ciampino. For Fiumicino, take the Leonardo Express train from Rome Termini (30 minutes), or the slower but cheaper Sabina-Fiumicino line that stops at every station. For Ciampino, take the train from Rome Termini to Ciampino train station, then a bus on to the airport.
If you choose the Stage 1 tour (Bourg St Pierre to St Martin): Reach Bourg St Pierre via Geneva Airport, with a direct train to Martigny and a bus from Martigny station to Bourg St Pierre (Trenitalia for schedules and fares). For your return from Pont St Martin, take a train towards Turin or Milan. Turin Porta Nuova station is around 1 hour 15 minutes away, and Milan Central station around 2.5 hours with several changes, with an airport bus or train shuttle from either city on to Milan Malpensa or Turin airport.
If you choose the Final Stage tour (Montefiascone to Rome): Fly into Rome Fiumicino Airport, then take a train to Viterbo (via Roma Trastevere station) and on to Montefiascone. The station is 5km from the town centre, so you'll need a taxi, or you can take a bus from Viterbo into the centre of Montefiascone to avoid the taxi fare. You could also fly into Rome Ciampino, though it's served by fewer airlines (Trenitalia for schedules). For your return, it's best to fly out of Rome via Fiumicino (Leonardo Express train from Rome Termini, 30 minutes, or the cheaper Sabina-Fiumicino line) or Ciampino (train from Rome Termini to Ciampino station, then a bus to the airport).
If you choose the Stage 4 tour (Fidenza to Aulla): The best airports for Fidenza are Parma, Bologna or Milan Linate. From any of these, take a train to Fidenza, where your hosts will pick you up, as the first accommodation is outside the town (Trenitalia for schedules and fares). For your return from Aulla, the best airport is Genoa. Take a train to Pisa (via Pisa Centrale station), which takes between 1.5 and 2 hours, or for Genoa, a direct train from Aulla to Genoa train station followed by a 30-minute shuttle bus to Genoa Airport.
If you choose the Stage 5 tour (Aulla to San Miniato): The best airport for Aulla is Pisa, served by many airlines from the UK and further afield. From Pisa Airport, take the Pisa Mover into the city, then a regional train to Aulla (around 1-2 hours). You can also fly into Genoa, though you'll change trains more often (2-3 hours). Trenitalia has schedules and fares. San Miniato sits between Pisa and Florence, so both are good airports for your return, with Florence around 1-2 hours away by train.
If you choose the Stage 6 tour (San Miniato to Buonconvento): San Miniato has its own station on the Florence/Pisa line, so fly into Pisa and take a train (1-2 hours, Trenitalia for fares and schedules). For your return, trains run from Buonconvento to Pisa (around 2 hours 20 minutes, changing in Empoli and Pisa) for Pisa Airport, to Florence Airport (around 2 hours 30 minutes, with a couple of changes), or to Rome Airport (around 4 hours 15 minutes, with a few changes depending on the time of day).
If you choose the Stage 7 tour (San Quirico to Viterbo): Since Buonconvento sits roughly equidistant between Pisa/Florence and Rome, you can fly into either. From Rome Fiumicino, take a train (via central Rome) to Chiusi, then a taxi to Buonconvento (around €50), or fly into Pisa and take a train to Siena, then on to Buonconvento (Trenitalia for fares and schedules). For your return from Montefiascone, take a train to Viterbo and on to Rome, then fly home from Rome Fiumicino Airport.