The course will never appear in preset nav, that’s one of the oddities of nav-to-nav transfer (the name for FMS to ILS auto-transition), and why I recommend setting it manually if radar vectors are expected. I found this weird too, and not how I would have designed it, but it’s just how the Challenger is.
When nav-to-nav transfer is able to work, you’ll see blue indications of localiser and glideslope deviation, both as blue diamonds on the deviation scales in the attitude/svs part of the PFD, as well as the CDI at the bottom with blue dashed “ghost” needles.
You should have FMS1 as the nav source on the left side, and FMS2 on the right side. Make sure baro settings are sync’d too.
Press the approach mode on the autopilot and you should see white LOC1 and GS indications on the FMA strip at the top of the PFD. Even then, it will hang onto FMS nav for as long as it can, until nearly established on the final approach track, then it will swap to LOC navigation. If you’re not on the glideslope, you may need to adjust VS or pitch to capture the glide properly.
Normal pilots rules apply : if it’s not doing what you expect, don’t just sit there, make it do it! Change nav source manually and fly the ILS. This is why, personally, I always set the course on preset nav as part of approach preparation.
Hope that makes sense.