Precision 7510 won't do 3 external monitors

Your issue may be with the Intel integrated graphics.  By default all video output is routed through and handled by the Intel graphics.

Go into your Bios and  go to the Graphics options.
Tick the checkbox for "Discrete Graphics Controller Direct Output Mode"  This will route all external displays directly through the Nvidia GPU.

The other option would be to un-check the "Enable Switchable Graphics" option.  This will allow your Nvidia Quadro to be the only GPU that the computer uses. (I am using this option for my Precision 7710)

Thanks for the suggestion. Unfortunately when I tried it, I still have the same problem. When the HDMI cable is attached, no video is going through the USB-C cable to the WD15 dock so I lose the 2 FHD monitors attached to the dock and only have the monitor attached to my 7510 via HDMI. 

Have you tried using the built in mini Display-Port instead of the HDMI? Or the mini Display-Port instead of one of the TB Dock video ports?

Good thought, forgot top try that. So, just tried the mini-DP instead of HDMI and the monitor never registered input while the laptop was connected to the WD15. So to recap, the 2 FHD monitors hooked up to the WD15 dock work through the USB-C connection but if I plug in an HDMI, then I lose the WD15 monitors. When I plug it in to the mini-DP, I still have the 2 WD15 monitors but nothing is getting to the 3rd. Just seems odd that a basic Latitude 5400 with integrated graphics can do the 2 monitors through USB-C to the WD15 AND the 3rd monitor through HDMI plugged directly into the laptop, but a Precision 7510 can't.

One of your problems may have to do with the age of your percision laptop.

The Precision 7510 uses 6th Generation Core CPUs with Intel HD Graphics
The Latitude 5400 uses 8th Generation Core CPUs with Intel UHD Graphics

How about trying to use both the Mini DP and HDMI from the Laptop and only have the dock run one monitor?

This works for me (4 monitors w/ 7510 laptop)

I am using a Dell 7510 laptop with M1000M quadro card.  I am running 4 monitors (laptop monitor, samsung 4K 43" tv through the laptop HDMI, and quantity 2 Dell U2410 1920x1200 through laptop mini display port with a Monoprice 121974 2-Port Displayport 1.2 to HDMI multi-stream transport Hub (DP to HDMI).  All 4 monitors are at native resolution and are working without issues.  The display port hub is essentially splitting the mini display port into 2 HD monitors (1920x1200).  I did not have to do any bios changes or anything special.

I hope this helps someone else get this set up?

@Berean  It sounds like the USB-C and HDMI outputs might be fed by a common GPU output within the system, which might explain why those specific ports can't be used together.  In terms of USB-C and MiniDP, try having only one display connected to the WD15.  Does that allow the display on MiniDP to light up?  If so, have you been keeping the built-in display active during all of this?  In that case, you'd essentially be trying to run four displays, and even the latest Intel GPUs don't support more than 3 simultaneous independent displays, and even though you have a Quadro GPU, systems that have discrete GPUs still often have most or all of the display outputs wired to the Intel GPU, with the discrete GPU running as a render-only device when needed.  NVIDIA calls this technology NVIDIA Optimus.  So that could be the issue here.  If having a single WD15 display and a MiniDP display works, try disabling the built-in display and seeing if you can get your two WD15 displays plus your MiniDP display all running at the same time.

Then again, based on the note from @HammyBone  about running 4 displays from this system, it's possible that at least certain outputs are wired directly to the NVIDIA GPU, in which case displays attached that way wouldn't count toward the Intel GPU max.  If you want to see which GPU controls which outputs, open NVIDIA Control Panel and go to the PhysX Configuration section.  In there you'll see a diagram showing which GPU controls each active display.  Connect a display to any output you want to test and see what it indicates.

@Berean  Sorry, when writing my reply above I missed the note that you don't have the built-in display active.  But just to confirm, are you running Windows rather than Linux?  The reason I ask is that on Linux, the built-in display can't be fully disabled, so even if you're not using it, it always counts as an active display and therefore counts toward the 3 display max of Intel GPUs.

Thanks for the replies. I have come to the conclusion that I can't do what I want with the equipment I have. I was running the 2 FHD monitors through the dock so I could just swap between my work and personal systems easily. When I got the QHD monitor from my office (part of work from home move) I was hoping to be able to have a 3 monitor set up by just plugging my Precision into the dock (as I have for the 2 FHDs) and the HDMI cable as I have with the Latitude. Too bad I can't get a work system with Thunderbolt then I could just upgrade to a TB dock and problem would be solved.