There are shades of grey here. VM's thick coax cable is triple shielded using (TTMOMK) aluminium foil and aluminium braid. I have seen various designations for the cable model number. I suspect this is something made to order for them and not available for general sale. That's understandable; first, they want to keep the cost down hence using aluminium rather than copper which is superior but much more expensive. Second, my guess is several manufacturers bid for supply contracts on price, hence the different designations, so supply swaps around from time to time. Third, if the cable can't be bought retail then it effectively resolves any customer situations where a householder says "I used the same cable you do". If it isn't available then VM can legitimately say that whatever the cable is, it doesn't meet their specifications.
A cable of similar construction that is available retail is Webro HD100. In fact the Webro site calls it
HD100 Ultra. It is triple shielded, and because it uses copper rather than aluminium has better performance than Virgin's aluminium-based cable. If there's going to be an issue with the VM engineer refusing to hook up a user-installed cable then it centres on attenuation caused by the cable being mishandled.
When coax is kinked or bent through too sharp an angle, or if it is stood upon or crushed, then it changes the relationship between the conductor, foam dielectric insulation, shielding and outer. These pinch points cause the signal level to drop (attenuation) and that upsets the noise margin within the network. This goes for any install using coax be it Freeview, satellite or cable, but cable is particularly sensitive to this because the cable is attached directly to the rest of the network for the other houses in the street, and the data cabinet etc whereas satellite and Freeview are physically isolated by the dish or aerial. IOW, Sky's satellite doesn't stop working just because Mr Jones at No.37 bent the cable a bit too far
So, in order for a VM engineer to hook up to a customer-installed cable then it has to be tested first to assess if it is damaged. There are line testers that can measure and assess any attenuation, and I would expect that that is something a VM engineer has to do anyway since VMs network runs on fairly tightly controlled signal to noise ratios. If the VM engineer cocks up when installing cable themselves then they'll know where and can splice in some new cable to replace the damage. If the engineer is faced with a customer's own handy work and no idea how well or poorly it has been done then it can become a bit of a stand off; you can imagine how that conversation might go.
The best thing I can suggest then is to have a look at the
VM Forum and take some advice from those closer to the source. The other thing to mention is that each Tivo style VM box will need a network connection back to the VM router/hub too. The boxes uses a wired connection (Ethernet port) so you need to think about running Cat5e or Cat6 along with the coax.