My post primarily answers the titled question such that the answer works for people that have read and have not read Mahou Sensei Negima (MSN).
Part 1 (For people who have read MSN. If you have not read MSN, skip to part 2)
Remember Eva from MSN? She is the same person as Yukihime-sensei. Over the years, Eva figured out a way to change her appearance so she could live life more normally - as an adult.
Basically, everyone from MSN presumably died (since ~100 years have passed) except for Eva since she is an immortal vampire.
- Page 1 - Eva gives her introduction
- Page 2 - She shows how the people from MSN are gone now
- Page 5 - Shows her current UQ team
- Page 70 - Eva reveals her true identity to Touta
- Page 16 - Eva reverts back to her 10 year old true form. She stays this way until page 38, where she returns back to her older fabricated form.
Both UQ and MSN are related through Eva, although MSN followed Negi's journey (main character of MSN). UQ's story is being told by Eva, who is reflecting on the past. Eva plays an important character in both series, as her role is to train the main characters, Negi and Touta.
Also note that Touta Konoe is Negi Springfield's grandson (Ch 1 Pg 18). Given that his surname is Konoe, we can also assume he is related to Konoka Konoe from MSN as well.
While UQ Holder is the sequel to MSN, there is nothing critical that you have missed. In other words, you can read and understand UQ Holder perfectly fine without reading its prequel. There are the occasional references to MSN though, but they do not affect the plot in any way.
A good analogy for this would be the difference between Dragon Ball and Dragon Ball Z. You can pick up Dragon Ball Z and fully understand everything without watching Dragon Ball. And in this case, UQ Holder and MSN have a time skip of 3 generations later, so every character except Yukihime is presumed dead now.