Waterdeep hasn't been examined in great detail by any 5th edition material, but resources from previous editions should still be useful in understanding the city demographics. The 3e sourcebook City of Splendors: Waterdeep features (from page 33 on) a description of the various armed forces associated with the city:

The figures given in this book largely agree with those suggested by the 2e City of Splendors, albeit that the prior book also notes that:

So it could easily be the case that these organisations are actually somewhat larger than is widely believed, thanks to the shenanigans of the Lords.

Based on 3e material, Waterdeep has standing armed forces of nearly 5,000 people

Waterdeep hasn't been examined in great detail by any 5th edition material, but there is plenty to draw on from previous editions. The 3e Forgotten Realms Campaign Setting states that the City of Waterdeep has a population of ~133,000 people, and the wider Waterdhavian region of control/influence extends 30-40 miles into the countryside and totals ~1.35M people. The more specifically focused City of Splendors: Waterdeep features (from page 33 on) a description of the various armed forces associated with the city:

The figures given in this book largely agree with those suggested by the 2e City of Splendors, albeit that the prior book also notes that the size of these forces might actually be considerably larger than commonly believed:

Another thing to keep in mind when referring to these resources is that, in the canonical FR timeline, there's a gap of just over a hundred years between the 3rd edition material and the 5th edition (1/2/3e covering the rough period 1357DR to 1376DR, 4e starting at 1385DR but mostly set in 1479-1486DR, and 5e play set beginning in 1489DR), and a lot of things have happened to Waterdeep and Faerun in that time - the Lord Piergeiron mentioned above, for instance, is long dead. The governance of the city still seems to run on roughly the same lines, though, with the Masked Lords and Guilds in power, so the arrangement of military forces is likely quite similar. If it transpires that the population of the region has significantly changed, the numbers for Waterdeep's military forces can probably be proportionally adjusted to match.

The 3e sourcebook City of Splendors: Waterdeep features (from page 33 on) a description of the various armed forces associated with the city:

The Lords of Waterdeep enforce their rule and maintain peace and security through the city’s armed forces. Waterdeep’s defenders are divided into four branches: the Griffon Cavalry, the City Guard, the City Watch, and the Navy. The Gray Hands act as an elite company, called in for emergencies.

• The griffon cavalry number just a hundred members, and by class mostly aristocrats, fighters, paladins, rangers and warriors (aristocrat and warrior being less powerful NPC classes).

• The City Guard are given as numbering 1,200, but the texts notes that the membership can swell to as much as 12,000 in war-time, the numbers being bolstered by hiring mercenaries. The associated classes of the organisation are barbarian, cleric, fighter, paladin, ranger, rogue, and warrior.

• The City Watch number 1,600 members, including the classes of cleric, expert (another NPC class), fighter, paladin, ranger, rogue, sorcerer, warrior, and wizard. Specifically regarding your interest in the magic-users, the Watch has one squadron of a half-dozen wizards/sorcerers led by a "Senior Watch-Wizard" per ward, of which there are 7 (or 8, if the waters of the harbour also count) - so there's a low proportion of arcane casters in the membership, even allowing for some number outside those squads - probably no more than 5% at best. It's not clear what the distribution of divine casters such as clerics, paladins and rangers is.

• The City Navy has a standing membership of 2,000, but as the Guard, in wartime the ranks can swell to as much as 9,000 by the addition of merchant sailors. The associated classes are barbarian, bard, cleric, expert, fighter, ranger, rogue, and warrior.

Waterdeep hasn't been examined in great detail by any 5th edition material, but resources from previous editions should still be useful in understanding the city demographics. The 3e sourcebook City of Splendors: Waterdeep features (from page 33 on) a description of the various armed forces associated with the city:

The Lords of Waterdeep enforce their rule and maintain peace and security through the city’s armed forces. Waterdeep’s defenders are divided into four branches: the Griffon Cavalry, the City Guard, the City Watch, and the Navy. The Gray Hands act as an elite company, called in for emergencies.

• The griffon cavalry number just a hundred members, and by class are aristocrats, fighters, paladins, rangers and warriors (aristocrat and warrior being less powerful NPC classes).

• The City Guard are given as numbering 1,200, but the texts notes that the membership can swell to as much as 12,000 in war-time, the numbers being bolstered by hiring mercenaries. The associated classes of the organisation are barbarian, cleric, fighter, paladin, ranger, rogue, and warrior.

• The City Watch number 1,600 members, including the classes of cleric, expert (another NPC class), fighter, paladin, ranger, rogue, sorcerer, warrior, and wizard. Specifically regarding your interest in the magic-users, the Watch has one squadron of a half-dozen wizards/sorcerers led by a "Senior Watch-Wizard" per ward, of which there are 7 (or 8, if the waters of the harbour also count) - so there's a low proportion of arcane casters in the membership, even allowing for some number outside those squads - probably no more than 5% at best. It's not clear what the distribution of divine casters such as clerics, paladins and rangers is.

• The City Navy has a standing membership of 2,000, but as the Guard, in wartime the ranks can swell to as much as 9,000 by the addition of merchant sailors. The associated classes are barbarian, bard, cleric, expert, fighter, ranger, rogue, and warrior.

• The Gray Hands are basically a loose group of adventurers who have agreed to be on call in the event of emergency as opposed to a proper standing military. There's about 20 of them and they should all be unique, powerful adventurers.

The book suggests that associated classes for the organisations it describes are listed in order of prevalence, but by inspection every class list was given in alphabetical order, so you probably should ignore that and not assume that Waterdeep is 50% barbarians.

The figures given in this book largely agree with those suggested by the 2e City of Splendors, albeit that the prior book also notes that:

The size of the guard and the watch is known only to Piergeiron and the Lords, but is strictly controlled, and thought to be approximately 1,200 guard and 1,600 watch. In times of strife, Waterdeep usually hires mercenaries and installs guardsmen as officers over them. The professionally curious are warned that Piergeiron has deliberately subdivided the payrolls of these forces so that it is difficult for visiting diplomats and others engaged in snooping (pardon, sightseeing) to discover the true size and names of the guard and the watch, and these figures may only be two-thirds or even less of the true totals.

So it could easily be the case that these organisations are actually somewhat larger than is widely believed, thanks to the shenanigans of the Lords.