Yes. As long as you’ve connected your Nintendo account to your Epic Games account, any Battle Royale purchased content and progression you’ve acquired (except V-Bucks) is accessible across Switch, Xbox, Playstation, PC, and Mobile.
Since you are already playing on Switch, you will need to register full account details before accessing the progress on Xbox, Playstation, PC, and Mobile.
In order to register full account details, complete the steps outlined in this article
under If you have played Fortnite on Switch already.
NOTE: When registering account details, you will need to use an email address that is not already registered with another Epic Games account.