I would check that your adapter seats properly in the new multifunction hotshoe so that the sync pin on the bottom is actually touching the sync contact (the big one in the center of the "square". The new physical shape of the multifunction hotshoe with additional electronic contacts in the back may not be compatible with the adapter you have.
As far as I know, the new multi-function hotshoe on the R10 should be compatible with any flash gear that worked on the older traditional Canon hotshoe of your 400D, but this is still new territory as the multi-function hotshoe only came out in 2021 on the R3, and folks only really got their hands on multifunction hotshoes starting in 2022 and 2023 bodies (R7, R10, R8, R6ii).
I would also check that you know the following things are good and work:
- Flash (test button)
- Batteries in flash (test cycle on a smart charger)
- Cable (swap cables)
- Adapter (try on another camera; or possibly tape over the small four non-sync contacts on the hotshoe and mount the flash directly on the hotshoe.
- That the R10's flash setting isn't turned off.
- That the R10's pop-up flash is not engaged and the hotshoe's microswitch is not broken or stuck.