Yes there is an electrified link from HS1 to the WCML using the NLL.
Also one from the international platforms at St Pancras.
As far as I know they have never been used, certainly not by passenger trains.
Theoretically, a Eurostar (373) set could go King's Cross-Edinburgh-Falkirk-Motherwell-Manchester-Birmingham-Willesden-NLL-HS1-Channel Tunnel and on to the limit of Eurostar capability in Europe.
It would run into trouble with the German/Swiss/Italian systems, so maybe Menton on the Riviera would be the practical limit.
That's assuming clearance is available for the full length of the ECML/WCML which might be tricky.
Testing on the WCML of the NoL sets only covered Manchester-Bletchley, and ECML services were limited to King's Cross-York/Leeds.
Out of interest, the GWML electrification is still isolated from the WCML/ECML system, with a short gap at Acton-Acton Wells.