On the mobile version of Home at least (which is what I use) it doesn't show abilities for Pokemon from PLA; I assume it would if they were transferred to a different game.
Messing around some more, just dropping 60 random pokemon that have never touched home before...
Switch Home has no issues with this. Like I said before you can just plop them into Home, never moving them anywhere else, and so long as you're not viewing in LA Mode* it will display the proper data. But with this random Bronzor for example, it must be moved into some game (I used SWSH) to display the ability in Mobile Home; but the 59 other Pokemon won't show anything.
But wait, there's more! Specifically it is only showing the SWSH moveset. So with that in mind I did another experiment: I took that same Bronzor and a random Alola-origin Rowlet, put them back into LA, took them back out and...no longer displays the ability for either of them, and displays their LA moveset.
So that's why. Mobile Home only displays the latest info stored associated with that Pokemon. Switch Home just shows whatever matches the view mode and updates accordingly. Barring the next update "fixing" that you will never be able to see those Pokemon's abilities!
SIDE NOTE: god i hate switching between switch & mobile. You can only be logged into the account in one or the other, not both, so you have to save & exit out of the Switch version, close it, then open the mobile version and mess around there, then close the app, back to switch, etc.
*In Switch Home, when viewing Pokemon, you choose to access one of your games where it will display all the Pokemon in that game's style (ie: LA will see all the Pokemon with no abilities, a moveset that matches their level up moveset in LA if applicable, all balls are Strange, etc) or you can view it in "Home Mode" where it just lets you look at the Home boxes without bothering with another game. Judging by displaying a Pokeball and the default level up moveset from SWSH (showing blanks if they aren't in SWSH), I assume Home Mode is basically viewing it in SWSH mode as a default.