Ok so according to the site I posted it states that Gamma Hydrae II / IV is approximately 132 light years away from Earth.
Now it says :
Earth Gamma Hydrae II, IV (Neutral Zone) 132.111 ly TOS  The
Deadly Years, ST:TWOK (real star)
This neutral zone is actually not specifically the romulan one but a portion of space near the romulan and klingon one, we can most likely round off a few lightyears more and say it's about 135 lightyears... Another person on a different site said something similar :
No specific distance is ever (thankfully) given in all of Trek over
how far it is from Earth the Romulan Neutral Zone (RNZ). However in
ST:ENT's second season episodes "Minefield" and "Dead Stop", we learn
that the NX-01 is some 130 light years from Earth when it first
encounters Romulans. We might assume for the sake of being
conservative, that the Romulus-Remus star system is only a few light
years further beyond this point. Round up to 135 ly.
Now given that we are using "estimations" this is by far not a canon answer since I haven't found any legit distance between Earth and the Neutral Zone.
The ship in First Contact was USS-Enterprise-E and as such is capable of going almost warp 10.
Warp 8 is the highest speed a Sovereign-class ship was known to have
traveled on-screen. According to Star Trek: Starship Spotter, the
maximum warp of the Sovereign-class ship was warp factor 9.7. However,
Star Trek Evolutions gives the Sovereign a maximum warp of 9.985.
If we use this information from Warp Factor it states that at Warp 9.975 40 light years can be done in 5 days :
9.975 2,922 40 light years 5 days VOY: "Relativity"
Now we can divide the total light years 135 divided by 40 and then multiplied by the amount of days. It should take approximately 16.875 days to get from the Romulan Neutral Zone to Earth if we assume that it is 135 light years away.