Immediately? You need to find some other way to prove you're still you. Many sites which enable WebAuthn have a pile of single use random text codes you can use to do this, write at least a few down somewhere safe.
The specification for WebAuthn (and presumably U2F but that's legacy and shouldn't be used for green field deployments) explicitly tells Relying Parties (that'd be the web site you're enrolling with) to allow multiple authenticators to be enrolled†
They are keys after all, so it probably feels reasonable to have a key you're carrying and one spare at home. Maybe if you lose your keys a lot, buy three just in case.
Unless you've got a FIDO2 device (not just FIDO) enrolled for usernameless authentication, the device doesn't even know who you are. So if you lost it on the train, or at a crowded event, relax, even if somebody found it intact and is curious the device can't help them log in as you since it doesn't even tell its new owner who you are. In fact it actually works perfectly well for them too, to secure their Facebook or whatever, in this way it's more like if you lose a quarter than if you lose your car keys.
† Now somebody will point out that AWS doesn't do this, and somehow this fact will be a justification for why an entire technology is bad, rather than yet another shortcoming of AWS...
