Hello
@plake.
Welcome to the forum.
I don't have recent knowledge, [I took the bus in 2018] but the bus from Aosta to the pass generally runs in the summer. During the winter there is a bus that goes
through the tunnel during the summer there are two bues to and from the hospice, Swiss on the Swiss side and Italian on the Italian side. You change, with a pause, at the pass, which is quite a tourist destination during the season.
This is a reply I got from the ever-helpful tourist office in Aosta earlier in the year. They are always very responsive to enquiries
aosta@turismo.vda.it. The website
https://www.lovevda.it/en
The bus Aosta-Hospice Gran San Bernardo will be active again from mid-June. The timetable is not yet present on the site, it will be loaded later at the following link https://aosta.arriva.it/en/timetables-and-routes/ Normally there are two runs per day (one in the morning, the other in the afternoon) from Monday to Saturday.
As for the Orsières-Le Grand St.-Bernard line, you should consult the website of the Swiss bus company https://www.tmrsa.ch/en/
i don't think I can see the timetable for the extension from St Rhémy to GSB at the moment. If I were you I would check with the tourist board.