Nerull was a jerk who wanted to be king of the gods. The other deities were happy when the Raven Queen croaked him—happy enough to raise her to godhood in his place—but didn't want a repeat performance. So they tweaked her portfolio a little, and she later added a couple extra domains of her own.
To prevent her from becoming a tyrant in the same mold as Nerull, the deities elevated this mortal shade to their own ranks and appointed her the goddess of death—but not the dead. Although sovereignty over the dead has been denied her, over the long ages of her reign she has added winter and fate to her domain. [Manual of the Planes 111]
To the best of my knowledge no single god now has the curation of souls as their responsibility. The Raven Queen makes sure souls move on, but there are at least a half-dozen different gods and demons who oversee various afterlives which souls might find themselves in depending on their beliefs and behaviour in life. (Vecna's Undeath portfolio is about keeping souls out of the afterlife and defying Death, so that's not relevant.)
According to Netheril: Empire of Magic (N:EoM), at the time of the Netherese there were very few human deities and each one of them carried very many portfolios. It seems like as the millennia went by, those portfolios got shared by many new powers.
For Mystryl, the list went like this: Magic, spells, the weave, wizards, spellcasters, energy, creativity, knowledge, invention, song, time, spring (page 49 of The Winds of Netheril).
We can compare this list with the portfolios given in the FR Adventures (FRA) and Faiths & Avatars (F&A) sourcebooks. We see that every portfolio except Time is covered: Azuth (mages, energy according to FRA; wizards, spellcasters according to F&A), Oghma (knowledge according to FRA; knowledge, invention according to F&A), Lathander (spring according to FRA; spring, creativity according to F&A), Milil (song), and of course Mystra (magic, spells, the weave).
Given that every other one is somehow covered, and the great many commonalities between N:EoM and F&A, I would agree with Carcer's statement and say Time appears to have been intentionally left out by TSR. (F&A lists Julia Martin with Eric Boyd as designers. For N:EoM, Julia Martin is listed under "additional design" and Eric Boyd is listed as a proofreader.)
There is one final curiosity: Amaunator is mentioned (in both N:EoM and F&A) to have misinterpreted some godly contract and wanted to preside over passage of time, but decided not to interfere with Mystryl. It looks like after Mystryl's demise, he still did not claim Time. So perhaps Mystra could still be the "hidden inheritor" of the portfolio and might someday claim it openly. It might also be worth mentioning that three "time gates" that allow characters to travel back to the Netherese period were still active as of 1370DR, and they were being protected by Mystryl's magic (N:EoM). It would be an interesting scenario to have Midnight/Mystra sending an avatar/proxy to meet her old self, Mystryl.
On a related note: Here is a list of powers from F&A who grant major access to the sphere of Time to her/his speciality priests: Amaunator, Auril, Chauntea, Cyric, Eldath, Kelemvor, Leira, Mask, Mielikki, Myrkul, Mystra, Shar, Silvanus, Sune. We broadly see that they are mostly deities of nature, death, or deception. If you really want to assign the portfolio of Time in your own game, consider one of these.
Deities followed by chronomancers
There is an article with the title Chronomancy & the Multiverse, version 1.1 by Roger E. Moore, published as "an updated and expanded excerpt from the Chronomancer", a 2e accessory. The article was made public by the WotC on their free downloads for previous editions site. This article has a section of the effects of chronomancy in the Realms, and relates time with deities of knowledge and writing. It is stated:
Native chronomancers are usually devout followers of both a deity of magic (e.g., Mystra) and a deity of time or history (e.g., Deneir or Oghma), serving as historians and information collectors. Elven chronomancers of Labelas Enoreth are possible. Shou Lung, in Kara-Tur, might have a secret, officially approved group of chronomancers, probably Historians, working for the Emperor.
Best Answer
In the Forgotten Realms Campaign Guide (page 50) of the 4th edition one can read (emphasis mine)
Mystra seems to be the only one capable to rule the Weave, as reported here\$^1\$: once the goddess was assassinated, the Weave was out of control and then it could burst and the Spellplague took place.
Moreover, Mystra assassination was planned by Shar: the adventure The End and the Beginning\$^2\$ explains that (emphasis mine)
Hence, Cyric did not take Mystra's portfolio because she did not die, its essence was "just" shattered and some of its fragments were collected by Shar. Shar's plan was described in Forgotten Realms Campaign Guide (page 50), in The Shadow Weave infobox:
Moreover, as described here (and references therein) there were some other parts of Mystra's essence around the world, confirming that the goddess did not actually die.
There is a series of tweets by Ed Greenwood that adds more details and may add more support to the above interpretation. It seems that there are several Weave anchors (see also here), both places and living entities, that have the role to be a safe net for the Weave. By Ed's tweets (emphasis mine)
Per my reading, the existence of these Weave anchors added a further complication to Cyric's chance to take the goddess' portfolio.
After Mystra's assassination, Cyric was imprisoned in the Supreme Throne by Tyr, Lathander and Sune, see page 74 of the 4th edition of FRCG: its influence was hugely reduced, and this imprisonment does not seem to allow the god to take over the portfolio of another dead deity. Furthermore, Cyric is driving more and more insane as its confinement in the Supreme Throne goes on.
\$^1\$ A very intriguing aspect, but there is no reference for this.
\$^2\$ From the website: Living Forgotten Realms (LFR) was the flagship Dungeons & Dragons Organized Play campaign for 4th Edition D&D, officially sanctioned by Wizards of the Coast.