The City Domain is aimed at clerics in modern cities, so it's hardly suitable for Dragon Heist.
Regardless of which features remain, most of them will be near-useless in a classic setting such as the one from Dragon Heist. If you want a "middle-age-fantasy city domain", you'll mostly have to homebrew (although you can certainly take inspiration from the UA domain).
Either way, let's take a look at the features gained by the domain:
- Domain spells: >50% of the spells refer to modern technologies, like computers or firearms, and are therefore unsuited for a classic fantasy setting.
- Bonus cantrip: same thing.
- Bonus proficiencies: same thing, although "land vehicle proficiency" is also a thing in classic settings. But for real, how often do you use vehicles in your campaign? Mostly to travel between towns, in which case you can hire a guy. Unlike in modern times, you don't chase bad guys with your vehicle, since horses without a vehicle attached are simply a lot faster.
- Heart of the City: works fine.
- CD: Spirits of the City: not entirely useless, but flavorwise not particularly well-suited and not as effective as it would be in a modern city.
- Block Watch: works fine.
- Divine Strike: works fine.
- Express Transit: depends on if you have bus stops in Waterdeep (just kidding, it's useless)
As you can see, most features are useless in a classic fantasy setting like Waterdeep. Sure, some are valid, but I highly doubt that any of your players will want to use a domain that's unuseable 60% of the time.
If you do want to provide a "city domain" option for your players, I suggest you make a homebrew subclass. That goes along the lines of what you're already asking, but RPG SE doesn't design homebrew starting from zero. Once you've got a first draft, you can ask for help on improving it.