Partial answer...

Considering:

https://www.sciencedaily.com/releases/2018/01/180111223914.htm

>NASA engineers has demonstrated fully autonomous X-ray navigation in space

NICER/SEXTANT

https://www.asterlabs.com/publications/2006/Sheikh_et_al,_AIAA_JGCD_Jan_Feb_2006.pdf

>Spacecraft Navigation Using X-Ray Pulsars

China's XPNAV 1

https://space.stackexchange.com/q/24448/40489

https://space.stackexchange.com/q/942/40489

https://space.stackexchange.com/q/21336/40489

https://ntrs.nasa.gov/citations/20190000639

>Predicted Performance of an X-Ray Navigation System for Future Deep Space and Lunar Missions

https://physicsworld.com/a/x-ray-pulsars-plot-the-way-for-deep-space-gps/

>X-ray pulsars plot the way for deep-space GPS

https://www.wired.com/story/nasa-just-proved-it-can-navigate-space-using-pulsars-where-to-now/

>NASA Just Proved It Can Navigate Space Using Pulsars. Where to Now?

https://arxiv.org/ftp/arxiv/papers/1704/1704.03316.pdf

>Pulsar Positioning System

I had erroneously assumed that JWST would make use of XNAV. But no.

Maybe, probably, due to the time when the project was first started and the length of time it took to get to launch, XNAV probably wasn't matured enough to get on board. 

So instead there is this:

>**Navigation Concepts for the James Webb Space Telescope**

https://ntrs.nasa.gov/citations/20040082113

>The  Mission  Engineering  and  Systems  Analysis  Division  at  the  Goddard  Space  Flight  Center  (GSFC)  is  supporting  the  JWST project by developing navigation concepts that meet nominal orbit determination accuracy requirements on the order of  50  km  in  position  and  20  millimeters  per  second  (mm/s)  in  velocity  (3-sigma).  

>These  requirements  are challenging  because  of  the  unusually  large  solar  radiation  pressure  (SRP)  forces  that  will  be  experienced  by  the  spacecraft  and  the  frequent  attitude  reorientations  and  unbalanced  momentum  unloads  that  are  planned  for  this  mission.

>This paper evaluates ... approaches for meeting the navigation requirements:
 - **Ground  navigation**  using  **standard  range**  and/or  **Doppler  measurements**  from  the  **Deep  Space  Network**  (DSN), which is the *current mission baseline*   
 - Onboard navigation using the **communications hardware and Sun sensor** *baselined for this mission*  
 - Onboard navigation using **optical celestial navigation sensors** that can measure the angle between the Earth and the Moon or a star and the Moon.


>The ground navigation approach processes standard range and Doppler measurements from the *Deep Space  Network*.  

>The  onboard  navigation  approach  processes  celestial  object  measurements  and/or  ground-to-spacecraft  Doppler  measurements  to  autonomously  estimate  the  spacecraft’s  position  and  velocity  and  Doppler  reference frequency. 

>The  ground  navigation  approach  provides  stable  navigation  solutions  using  a  **tracking  schedule  of  one  30-minute  contact  per  day**.  

>The  onboard  navigation  approach  that  uses  only  optical  quality  celestial  object  measurements  provides stable autonomous navigation solutions. 

So, the answer seems to be a combined ground and onboard navigation approach making use of the DSN.

..and not GNSS 

So, whilst celestial GPS via Pulsars seems to be the way forward, they do not seem to be on JWST.. 

Further:


https://ntrs.nasa.gov/api/citations/20080030196/downloads/20080030196.pdf

>Ranging is required for JWST, using alternate ground stations in the southern and northern hemisphere.

>The NASA Spectrum office objected to provide more than the 10
MHz band in X-band range and suggested using Ka-band. JWST project decided to move to K-band and have one (1) 4-hour contact per day for communication and ranging. 

https://www.esa.int/Enabling_Support/Operations/ESA_Ground_Stations/ESA_to_pick_up_as_Webb_phones_home

>ESA to pick up as Webb phones home

https://www.esa.int/ESA_Multimedia/Images/2019/02/Ultra-precise_navigation

>Ultra-precise navigation

>"How not to lose a spacecraft"

[![enter image description here][1]][1]

https://www.esa.int/Enabling_Support/Preparing_for_the_Future/Discovery_and_Preparation/Deep_space_communication_and_navigation

https://journals.sagepub.com/doi/pdf/10.1177/002029400804100302

>For near-earth space explorers, a variety of autonomous navigation methods have been proposed and explored, including a magnetometer-based
navigation method, Global Position System (GPS), inter-satellite
link, and celestial navigation method. 

>However, for deep space explorers, celestial navigation is the only feasible way.


  [1]: https://i.sstatic.net/eqhEr.png