What are the lateral sensitivities of GPS approaches?
Check out AIM 1-1-9 for localizer information. Basically, the localizer is positioned so to have a course width of 700' at the threshold. So full deflection to either side is 350'. A CDI has 5 dots on each side, so at the threshold each dot is 70'. Near the Middle Marker, one dot means you’re off course by 150 ft. Near the Outer Marker, a one-dot deviation puts you about 500 ft. from the centerline. 

GPS (non-WAAS) approaches have a constant full scale deflection of .3 NM after the final approach fix.


  AIM 1−1−17 5 (e) (5) When within 2 NM of the Final Approach Waypoint (FAWP) with the
  approach mode armed, the approach mode will switch to active, which
  results in RAIM and CDI changing to approach sensitivity. Beginning 2
  NM prior to the FAWP, the full scale CDI sensitivity will smoothly
  change from ±1 NM to ±0.3 NM at the FAWP. As sensitivity changes from
  ±1 NM to ±0.3 NM approaching the FAWP,…


WAAS on the other hand is similar to the localizer.


  AIM 1-1-18 (d) 4 Both lateral and vertical scaling for the LNAV/VNAV and LPV approach procedures are different than the linear scaling of basic
  GPS. When the complete published procedure is flown, ±1 NM linear
  scaling is provided until two (2) NM prior to the FAF, where the
  sensitivity increases to be similar to the angular scaling of an ILS.
  There are two differ- ences in the WAAS scaling and ILS: 1) on long
  final approach segments, the initial scaling will be ±0.3 NM to
  achieve equivalent performance to GPS (and better than ILS, which is
  less sensitive far from the runway); 2) close to the runway threshold,
  the scaling changes to linear instead of continuing to become more
  sensitive. The width of the final approach course is tailored so that
  the total width is usually 700 feet at the runway threshold. Since the
  origin point of the lateral splay for the angular portion of the final
  is not fixed due to antenna placement like localizer, the splay angle
  can remain fixed, making a consistent width of final for aircraft
  being vectored onto the final approach course on different length
  runways. When the complete published procedure is not flown, and
  instead the aircraft needs to capture the extended final approach
  course similar to ILS, the vector to final (VTF) mode is used. Under
  VTF, the scaling is linear at ±1 NM until the point where the ILS
  angular splay reaches a width of ±1 NM regardless of the distance from
  the FAWP.