Okay, I just did some analysis work.

I had kernels 5.4.0-47 and 5.4.0-48 on my system and with both at first external monitors didn't appear to work.

Then I installed 5.4.0-45 and 5.4.0-42 as well, so I got four kernel versions on my system. Then I tried booting with each kernel version both with laptop lid closed and opened several times. These are the results.

--- SNIP ---

Kernel 5.4.0-42, lid open:
- LUKS authentication: displayed on internal and external screens
- login screen: displayed on internal and external screens
- Budgie desktop: only displayed on internal screen, external screens do not show up under display settings

Kernel 5.4.0-42, lid closed:
- crashes during LUKS pre-boot authentication

Kernel 5.4.0-45, lid open:
- LUKS authentication: displayed on internal and external screens
- login screen: displayed on internal and external screens
- Budgie desktop: only displayed on internal screen, external screens do not show up under display settings

Kernel 5.4.0-45, lid closed:
- LUKS authentication: shows up on external screens most of the time, sometimes crashes during pre-boot authentication
- Login screen: most of the time shows up on external screens, sometimes does not show up on external screens, but then shows up on internal screen after lid is opened
- Budgie desktop: most of the time only displayed on internal screen, external screens do not show up under display settings, rarely shows up on external screens, switching to internal screen after lid is opened

Kernel 5.4.0-47, lid open:
- LUKS authentication: only displayed on internal screen
- Login screen: displayed on internal and external screen
- Budgie desktop: only displayed on internal screen, no external screen recognized

Kernel 5.4.0-47, lid closed:
- LUKS authentication: shows up on external screens
- Login screen: shows up on external screens
- Budgie desktop: either does not show up on external screens and does not show on internal screen after laptop lid is opened, or shows up on external screens and switches to internal screen after laptop lid is opened

Kernel 5.4.0-48, lid open:
- LUKS authentication: only displayed on internal screen
- Login screen: displayed on internal and external screen
- Budgie desktop: only displayed on internal screen, no external screen recognized

Kernel 5.4.0-48, lid closed:
- LUKS authentication: shows up on external screens
- Login screen: shows up on external screens
- Budgie desktop: either does not show up on external screens and does not show on internal screen after laptop lid is opened, or shows up on external screens and switches to internal screen after laptop lid is opened

--- SNIP ---

Man, that's horrible. I somehow guess I could do this another hundred times and still get different results. It somehow appears like there's a race condition during device enumeration or something like that, which would be a kernel issue then, and will probably be horrible to trace down.

However, assuming that this is a problem with device enumeration, some behaviour appears odd. First, reconnecting the dock does not appear to help alleviate the issue, which is kinda weird if it's a problem with device enumeration, since I would expect the devices to get re-enumerated when they are removed and subsequently connected to the system again. Also, I wonder if the kernel actually re-enumerates devices when switching from pre-boot authentication through display manager to desktop manager, since it appears we can "lose" device detection at these points in time.