Sorry I’m a bit late to this Martyn, but to update. Skyworks is a great airline, very friendly with spotlessly clean aircraft. It’s often used by British Embassy staff as well as ministers and employees of the various Swiss ministries. They use the Saab turboprop which is very comfortable and has a full aisle of single seats when travelling alone. However when I flew the flight was only about 60% full so I moved over to the double seat.
It makes a stop in Basel and from memory it’s only about 15 minutes, and you can be pretty certain that those staying onboard are all (UK or CH) goverment employees!
When you leave Bern you will not need a taxi as the tram/bus service is excellent and drops you right in front of the station. If you’re staying the night (I can recommend some hotels/restaurants if you do) the hotel will usually give a free pass for public transport.
If going to Frankfurt you have one direct train at 1104 otherwise you have to change in Basel.
One word of advice, if you top up with snacks at Bern station, buy your water at the Co-op. 90 cents versus CHF 4.80 at the kiosk right next door.