I'm not sure that's correct. Memory is that it started as a (Western Region class 116) dmu service, and certainly I photographed such a working at Clapham Junction in September 1996. The critical bit would be Willesden Junction to Mitre Bridge Junction, which would not have been electrified as part of the Eurostar scheme.
When I was a lad on the relief mid-late 80's the West London line had 2 inter-City services per day; Manchester>Dover and return [1O66 up, can't remember the down headcode] and also a Glasgow & Edinburgh (join/split at Carstairs back then)>Brighton and return [1S76/1O90]. The Manchester travelled via the WCML 87 hauled: locos swapped at Mitre Bridge junction. The Brighton was class 47 hauled, turned off at North Pole junction and went via Reading. I expect they swapped the 47/87 at Birmingham New Street. All of these services stopped at Olympia, Sundays excepted.
Re. the wires as above; in fact the wires went as far as Mitre Bridge from Willesden low level, and that's where they swapped the leccys for diesels on the Manchester/Dover service.
There was the occasional charter to Olympia for Exhibitions etc., the odd 'footex' or 'mystex' and of course the premier peak hours 'Kenny Belle' service Olympia/Clapham Junction as mentioned in posts above. But the line was mainly used for freight.
I've always had a soft spot for 1S76 & 1O90 having signalled them through Olympia, North Pole, Annan (diverted), Dumfries (ditto), Edinburgh Waverley and Carlisle.....them were the days.