Getting text name of current project CRS in QGIS Layout

I think you will need a custom function to do this, the following seems to work though I haven't tested it extensively.

from qgis.core import *
from qgis.gui import *
from qgis.utils import iface

@qgsfunction(args='auto', group='Custom')
def getCrsDescription(feature, parent):
    """
    Returns the description text of the project CRS
    <h2>Example usage:</h2>
    <ul>
      <li>getCrsDescription() -> WGS 84</li>
      <li>getCrsDescription() -> OSGB 1936 / British National Grid</li>
    </ul>
    """
    return iface.mapCanvas().mapSettings().destinationCrs().description()

This behavior will be improved in an upcoming release of QGIS due to the below fix by Nyall Dawson and Alex Royas: https://github.com/qgis/QGIS/commit/a10bb387a767c21b27d345138227286ac2bd7ce2

New expression variables:

project_units

• Unit of the project's CRS

project_crs_description

• Name of the coordinate reference system of the project

project_crs_acronym

• Acronym of the coordinate reference system of the project

project_crs_ellipsoid

• Acronym of the ellipsoid of the coordinate reference system of the project

project_crs_proj4

• Proj4 definition of the coordinate reference system of the project

project_crs_wkt

• WKT definition of the coordinate reference system of the project