It looks like about 57 years.

Assuming constant acceleration to the 1/2-way point, flip, deceleration, and using http://www.projectrho.com/public_html/rocket/slowerlight3.ph... :

  Time elapsed (in starship's frame of reference, "Proper time")
  T = (c/a) * ArcCosh[a*d/(c^2) + 1] (given acceleration and distance)
  year = 365.25*24*3600; c = 3E8; a=9.8; d=1.25*1_000_000*(c * year)
  from math import acosh
  T = (c/a) * acosh(a*d/(c*c) + 1)
  print(T/year)
  => 14.3 years each quarter 
  => 57 years round trip

The speed at flip would be 99.99999999993978% c - good thing intergalactic space is mostly empty.

Bad news CMB shifts into infrared, visible, uv, xray and then hard gamma.

Oh, what a shame. I was about to install some rockets on my RV and head off to Andromeda. Guess I don't need to put my newspaper subscription on hold now.

since we are talking insane speeds and energies required to reach them why not just say you are in a bigger ship with more radiation shielding a dozen or so meters of water in the hull and a layer of lead a foot thick aught to stop most of it

FWIW, the Project Rho page I linked to also gives the derivation of the relativistic rocket equation. When I plugged in numbers, assuming an exhaust velocity of the speed of light (ie. impossibly high), I get a mass fraction of about 2,000,000 to reach top speed. Then another 2,000,000 to slow down.

Assuming the fully loaded RV weighs 5 tons, this means at max velocity the rest of the ship weighs 10 megatons, which is 10M cubic meters of water, or a cube 200m on each side.

That sounds like plenty of material, right?

The same Project Rho page links to https://arxiv.org/ftp/physics/papers/0610/0610030.pdf which calculates that at the relatively slow 0.995c "the penetration depth of protons of this energy will be ~40 m in water and ~10 m in titanium".

For 99.99999999993978% c, even 10M cubic meters isn't going to be enough.