All I know is that it was during the later years of the Eighth Doctor and caused him to regenerate into his ninth form.

It was not during the TV series. It happened sometime between "Doctor Who", the 1996 TV movie, & Series 1, Episode 1, "Rose". To judge by the (9th) Doctor's behaviour in "Rose", his regeneration was very recent at that point, so the end of the Time War must also have been recent. It probably wasn't immediately before his encounter with Rose Tyler, however. The Doctor has almost always suffered some kind of post-regeneration trauma & there was no sign of that, implying he'd had time to recover. He also spoke as if he'd been dealing with the Autons for a while -- days, at least -- before he rescued Rose from them.

To be correct, however, the series has never explicitly said that the Doctor regenerated because of the Time War. Had Doctor Who Magazine been allowed to regenerate the Eighth Doctor as had been planned, it would have occurred before the Time War. It has also been stated that the Time War referenced in the TV series is a different event than the Time War featured in the Eighth Doctor novels.