I have little knowledge in what is (for Europe; the OP is American) a highly specialised area of avionics, but there are two separate issues:
1) Getting an autopilot to track a GPS approach
laterally. This is easy. You traditionally do it in the NAV mode, and as one gets near, the full scale sensitivity automatically winds down from 5nm to 1.0nm to 0.3nm. If you have GPSS (a.k.a. roll steering) installed, the autopilot is either driven from a GPSS converter (whose output fakes the heading bug output of an HSI) and then the autopilot is used in the HDG mode, or it is driven directly digitally, using ARINC commands, from the GPS, in which case you use the NAV mode.
2) Getting an autopilot to track a GPS/LPV approach
vertically i.e. a virtual glideslope. This doesn't yet exist in Europe because it needs EGNOS and that is still (
AFAIK) buried in some c0ckups despite it being "certified for life critical uses" very recently. Exactly how you get a particular autopilot installation to track this virtual glideslope coming out of the GPS, is going to be rather installation specific. Ideally it will be again via digital ARINC commands, but more likely there will be some "fake" whereby the GPS outputs a glideslope-like signal and the autopilot is fooled into believing it is an ILS glideslope, with the appropriate sensitivity.
I think the autopilot will have to be in the APR mode for tracking an LPV approach; I don't think any autopilot will even look at any external VNAV signal when in the NAV or HDG modes.
But you do
not fly normal (non LPV) GPS approaches in the APR mode; they are flown in NAV or HDG. So I am baffled....
Wigglyamp here should know exactly how this stuff is hacked together - even if nobody will have tested it in Europe (well not overtly, anyway
). I know some people in Spain have been testing LPV approaches for years, with a GNS480 with its firmware hacked to use EGNOS despite the Test flag being active.