score:3
You always check in with the operating carrier of the first flight. In your case, that would be British Airways. This is true both for online check-in, as well as at the airport/baggage drop/etc.
What happens after that can vary.
Depending on the carriers involved, their policies, and their computer systems, the first carrier might be able to give you boarding passes for all flight, or they might only be able to give you one/some. Which flights they can check you in for might be the same online as at the airport, or they might be able to check you in for additional flights at the (first) airport.
In some cases, if the first carrier is not able to check you in/give you boarding passes for the subsequent flights you may be able to do that online with the operating carrier for that flight. For example, I recently flew with United Airlines connecting to Oman Air. United was unable to check me in for the connecting flight, but I was able to do it myself online (on the Oman Air website) without any issues.
In your specific case, all carriers are OneWorld airlines, so there is a good chance the initial carrier (BA) will be able to give you all boarding passes, but if not, simply go to the transfer desk in either Helsinki and/or Osaka and they will be able to give you the boarding passes for the connecting flights. You WILL be able to check your bags all the way to your final destination, although I suspect you will need to collect them in Osaka being your first point of entry into Japan.
Upvote:2
Always check in with the airline operating the initial flight in your itinerary. In this case, you'll check in with British Airways for your outbound flights, and with JAL for your inbound flights.
Whether you can check in to all flights on the itinerary will depend a little on the arrangements the airlines have with each other, but typically (and in my own experience), you can check into all flights with the initial checkin, although sometimes you may not be able to select a seat with codeshare airlines. I've also run into the issue where I had to print or get a boarding pass at a connecting airport.