Cork is one of the few coverings where even the manufacturers say that it is not suitable for underfloor heating. There may be exceptions. Your heating engineer is not wrong.
In practice, the effect then depends on thickness, manufacturer, and insulation value. It will work somehow, but it would have been better, for example, to also equip the wall or ceiling with underfloor heating.
What will happen: the flow temperature of the heating system must be higher than usual so that the cork rooms become warm. All other rooms must be throttled because otherwise they become too warm. This costs pump electricity and heating efficiency, especially if a heat pump is installed, which would be unfavorable. Living rooms (sun, usage) and bedrooms often have a slightly lower heating requirement, which could balance it out a bit. With a gas heating system, you could take the risk if you are attached to it; with a heat pump, I would definitely avoid cork.
Vinyl is warm underfoot and, when glued down, performs almost as well as tiles for underfloor heating. Even with floating installation of very well-conducting coverings (no wood components), I see higher return temperatures.
The ongoing additional heating costs are hard to quantify. Probably not more than 25% worst case.