I just installed that exact same pack today (first time getting Voice Attack up and running) and am also using the Warthog. There are, unfortunately, a number of crucial steps that the tutorial videos don't mention. Wasted hours trying to figure this out. So here goes:
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1. You'll need to find the "right" HCS profile. I found some that some worked, and some didn't. I currently use one of the basic profiles - one listed under the "basic" profile subfolder and I believe - working from memory - I even chose one that wasn't specifically for the Warthog. I wanted to keep things as simple as possible.
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2. If you set up a custom bind for your Warthog (i.e. customized your ED bindings), chances are high that the HCS voice pack won't work right out of the box with ED. It's an easy fix, though. What you need to do is to dig into the HCS folder (wherever it was installed) and look for HCS .BINDS files. I believe (working from memory), they may be in (or near) the Profiles folder. I picked the one called HCS_somethingorother_Warthog.BINDS (not an exact name - working from memory). You'll need to copy that bindings file into your C:\Users\USERNAME\AppData\Local\Frontier Developments\Elite Dangerous\Options\Bindings\ folder. (Note: AppData is hidden by default, so ensure you unhide all files in the C directory).
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3. Once you've found and copy/pasted the above file, open ED and you should see a drop down option in the Controls menu for the afore-mentioned control scheme. This should work with ASTRA out of the box, but if you've modified your control scheme, you may find that the default HCS controls are suboptimal. So what you'll need to do is to customize again, BUT only change the values in the second column in ED (the first column holds the key binds that Voice Attack uses). Because ED allows 2 "keys" per command, you should be able to replicate your old control scheme 100% (I managed to).
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4. Once you've got the new custom bindings file sorted, I'd recommend copy/pasting it somewhere safe as a back-up. ED will "forget" those bindings between launcher updates; happens frequently, unfortunately.
Good luck!