During the First Doctor's era, the Doctor's granddaughter Susan was left on 22nd Century Earth. There doesn't seem to be any evidence that the Time War affected Earth; as the Gelth said, the War was "invisible to more primitive species". So why did the Doctor appear to forget about her when he described himself as the last of the Time Lords? If she really was his granddaughter, then she would have been at least partly Gallifreyan; and given the physical damage Donna Noble experienced when she became the Doctor-Donna, it's possible that an only-part-Gallifreyan might not have been able to survive, implying she was fully Gallifreyan.
According to extended canon (in particular, the Big Finish audio drama All Hands on Deck), Susan was recruited into the Time War by the Time Lords despite the Doctor's best efforts to keep her away from it - so when "New Who" came about (i.e. the 9th Doctor onwards), she was as lost as the rest of her race, making the Doctor's claim of being last of the Time Lords still true. Her current whereabouts are unknown, but she may have returned along with Gallifrey in more recent seasons.
Carole Ann Ford appeared as Susan in the 20th and 30th anniversary specials "The Five Doctors" and "Dimensions in Time" (the former being canon but the latter much less so), and she was involved in some of the 50th anniversary pieces such as having a bit part in the docu-drama "An Adventure in Time and Space", so it's not impossible for her to return in the role, but whether it happens will be up to her and the new show-runners.
The Doctor claims he is the last of the Time Lords because he thinks he killed them all when he destroyed Gallifrey, however it has been shown at various times this is not the case. The Master/Missy is the most obvious proof of this.
I also don't know if it has ever been truly established that Susan was the Dr's genetic granddaughter.
Your question of what happened to Susan has been well answered by @Conman according to extended canon.
At this time only the extended canon of Big Finish productions have made any reference to Susan after The Dalek Invasion of Earth. Susan's appearance in The Five Doctors is of course not meant to be after The Doctor left her behind on Earth, but rather during her time travelling with him. Like all instances of The Doctor's timelines crossing, these are anomalies which, although real, cease to be part of all but the current Doctor's established timeline. This is established in The Day of The Doctor when Ten leaves Eleven behind and it is said that Ten will not remember it. Also, the canonical short Time Crash attempted to explain away the aged appearances of previous Doctors (obviously due to the actors having aged) whenever their paths cross by saying that being in same place "shorts out the time differential". We can only speculate how this could affect companions traveling with them.
As to your supplemental question of why the Doctor appeared to forget about her when he described himself as "the last of the Time Lords", I have two possible lines of reasoning:
1. Susan is a Gallifreyan, but not a "Time Lord"
Some argue there is a difference, and that Time Lords are just Gallifreyans in possession of a TARDIS. As Susan was left behind on Earth to live as a human and start a life with David, The Doctor may not have considered her to be a Timelord, regardless of whether she is a full Gallifreyan or not.
2. She really isn't fully Gallifreyan
You suggested that a "part-Gallifreyan might not have been able to survive", citing the example of Donna. However Donna was physically fully human having been born so. Her human brain was not capable of holding The Doctor's mind. On the other hand, The Doctor himself is supposedly half-human. This idea established in the Eighth Doctor's only TV appearance was not popular and has sometimes been dismissed; yet that pilot has been embraced as canonical by subsequent series, and Moffat toyed with the concept during the story arc involving Ashildr. A prophecy of a "hybrid" left some believing it could refer to The Doctor as a half-human-half-Gallifreyan, and when The Doctor confirmed it was "me" it was deliberately ambiguous as to whether or not he meant himself or Ashildr who was now known by the name "Me".
Here's some supposition on the possibility of why Susan could never be a Timelord. The 5th Doctor story The Two Doctors suggested that the secret of Time Travel in a TARDIS was connected to Timelord physiology contained in their eye. The Sontarans were unable to get their version of a TARDIS to work without The Doctor's eye. The idea that the eye of harmony on the 8th Doctor's TARDIS was symbiotically connected to The Doctor's own eye, as well as the idea that trainee Timelords had to stare into the vortex may have been attempts to build on that idea. It was when The Master saw The Doctor's eye that he realised he was half-human.
If TARDIS travel through time is only possible due to a symbiotic connection to Gallifreyan physiology then maybe hybrids like The Doctor, as well as any future children and grandchildren, may only be True Timelords if they inherit the correct genes from their Gallifreyan heritage. We do not know who Susan's mother or grandmother were - they may have been human too, further diluting Susan's Gallifreyan heritage.