I feel somewhat conflicted writing this because Bell's book inspired many people to become mathematicians, including some prominent ones. However, it is not a canonical introduction to the history of mathematics and mathematicians, "historians of mathematics tend to distrust the historical reliability of most of Bell’s accounts"(Leo Corry). We recently had a question What's the famous story about a mathematician who gave a talk without saying a word?, which illustrates why. He embellishes facts, retells anecdotes, and makes up some of his own. Even normally bland Wikipedia has some choice words for Men of Mathematics.
Siegmund-Schultze summarizes it well:
"E.T. Bell’s book on Men of Mathematics is known to be notoriously unreliable and sketchy, but it is a good read and can instill enthusiasm for mathematics in the beginner, not least because it aims to show that “mathematicians can be as human as anybody else” and because Bell wants to “lead up to some of the dominating ideas governing vast tracts of mathematics as it exists today”".
So Bell's books are in the genre of historical fiction (he also wrote science fiction), which in itself is valuable, but it should not be taken for history. If you like the genre some call Muir's Of Men and Numbers
"Bell's clone", although it still ends with Cantor. A much better book on 19th century is Yaglom's Felix Klein and Sophus Lie, which despite the title gives a panoramic view of 19th century mathematicians, and is more accurate. Like Bell Yaglom is a good writer, but he is captivating in ways different from Bell's and Muir's. More scholarly works on 19th century are Parshall's Sylvester and Wedderburn.
Researching mathematician's life is a lot of work, especially given the number of sources for recent times, and individual biographies written by original researchers are the way to go if you want real history. Something along the lines of Constance Reid's Hilbert, Courant and Julia. I am afraid there are no "canonical" compilations. By the way, instead of Hilbert Reid was supposed to write an "update" of Bell's Men of Mathematics, but changed her mind after traveling to Göttingen. In that she was encouraged by her sister, Julia Robinson, the heroine of the last book. Later she wrote a book about Bell himself, "more of a detective story than a true biography", uncovering some details that not even his family knew about. You may also go for first hand accounts, like Russell's Autobiography, touching on his interactions with early 20th century mathematicians, or deeply personal and candid Grothendieck's Récoltes et Semailles featuring French mathematicians of 1950-60s, but keep in mind that first hand accounts are by definition opinionated and biased.
Mathematics Unbound: The Evolution of an International Mathematical Research Community, 1800-1945 is a collection of 20 scholarly essays on the period.
James's Remarkable Mathematicians: From Euler to Von Neumann goes into 20th century, and is less anecdotal, but he is not as good a writer, and see linked Siegmund-Schultze's essay about his history. O’Shea's Poincare Conjecture (touches on many 20th century mathematicians) gets a nod despite "many minor factual mistakes", it also has further reading pointers. "O’Shea, unlike James, is a mathematician who uses the work of historians with profit, gives credit where credit is due, immerses himself in the spirit of the times, and, crucially, imparts his mathematical knowledge in a form which the public can understand." MacTutor History has chronologically indexed biographies "useful as a first orientation for further research. Unlike in James’s biographies, there is also some mathematical literature for further reading added, although the manner of quoting within the biographical entries and the reliability of information provided are often similarly dubious".
Cambridge's Mathematical Reading List has a history section with brief characterizations of some mathematicians' biographies.
There is also an assorted collection of modern anecdotes on Math SE, and this unsigned Greatest Mathematicians of 20th Century, read at your own risk.
