When you've booked the flights with one airline, you don't have to worry about missing your connecting flight, also the chances of actually missing your connecting flight due to delays are a lot less compared to booking the flights separately. The gate staff will do everything they can to make sure connecting passengers will be able to board the plane. This is because there are financial consequences for the airline if you have to be re-booked on another flight.
You're already going to be checked in on the connecting flight when your still on your way. If your plane is delayed, the gate staff of the connecting flight will be notified. It's quite common for the connecting flight to be slightly delayed to allow passengers from connecting flights to board the plane. The gate will be left open until you arrive, provided the delay isn't too large. In airports that are much larger than Helsinki airport, the gate staff will sometimes deploy cars that are normally used for handicapped passengers to bring older passengers who cannot walk fast to their connecting flights in time.
If the delay is too large then you'll be either re-booked to another flight when you're still underway, or you'll be asked to go to the transfer desk where the re-booking will be sorted out. I've experienced such delayed quite a few times myself. I missed the connecting flight due to a large delay and I was re-booked during my flight. I had to print a boarding pass on arrival, and I saw that the boarding pass said that my new flight was due to leave 5 minutes ago, boarding had started 35 minutes ago when I was still in the air. But I could still walk to the gate, enter the plane and go to my seat.
If there are no connecting flights available that leave within a few hours, then you're going to be asked to go to the transfer desk. This gives passengers the option to not travel to their destination due to the delay, or pick another flight than the next one available. Depending on the delay, you'll be given meal vouchers, a hotel to spend the night in, and clothing, pajamas etc. as you won't have access to your baggage.