I think the easiest answer is "in the same direction that the north galactic pole (NGP) is from the north celestial pole (NCP)."
Which makes it easy to see that the answer is simply the right ascension of the NGP in celestial coordinates, which was defined as 12h 49m in B1950 coordinates, was 12h 51m in J2000 coordinates, and is currently about 12h 53m (mid 2024). See https://en.wikipedia.org/wiki/Galactic_plane
Note that in mid-2024 coordinates, the maximum angle between the ecliptic plane and the galactic plane (the inclination of the galactic plane) is just over 63 degrees: 60.0046 degrees, since the current precessed position of the NGP has Declination 26.9954.
ChatGPT offers this code to precess the NGP to the date given in the code (2024-07-01). (I've tested that it properly reproduces the J2000 coordinates, and it looks good otherwise):
from astropy.coordinates import SkyCoord, FK4, FK5
from astropy.time import Time
import astropy.units as u
# Define the coordinates of the North Galactic Pole in B1950
ngp_ra_b1950 = 192.25 * u.deg
ngp_dec_b1950 = 27.4 * u.deg
# Create a SkyCoord object for the NGP in the B1950 frame
ngp_b1950 = SkyCoord(ra=ngp_ra_b1950, dec=ngp_dec_b1950, frame='fk4', equinox='B1950.0')
# Define the target date for precession
date_mid_2024 = Time('2024-07-01')
# Transform the coordinates to the target date in the FK5 (J2000.0) frame
ngp_2024 = ngp_b1950.transform_to(FK5(equinox=date_mid_2024))
# Print the precessed coordinates
print(f"RA (2024.5): {ngp_2024.ra.to_string(unit=u.hr, decimal=True)}")
print(f"RA (2024.5): {ngp_2024.ra.to_string(unit=u.hourangle, sep=':')}")
print(f"Dec (2024.5): {ngp_2024.dec.to_string(unit=u.deg, decimal=True)}")