If an ILS fails the missed approach may be predicated on VOR. So, no big deal.
LPV is no different than any other RNAV instrument approach procedure (IAP) when it comes to a GPS failure. Guidance is gone (unless the aircraft is a high-end bird with IRUs). If radar services are available they are the backup. In a non-radar area the pilot has no choice but to dead reckon along the missed approach course (or courses) to the best of his ability. At some point VOR reception should occur (well, at least for now).
Pre-SBAS RNAV IAPs have been in wide use in the U.S. for 12 years, or so, with no loss of GPS yet, at least none that has been made public.
RNP AR is the exception, at least where RNP of less than 1.0 is used for the missed approach. In that case, the aircraft must have at least 1 IRU because the containment areas are too narrow for dead reckoning (or even radar vectors) in the event of a loss of GPS.