Main answer
Westcott and Hort influenced modern Bibles because their Greek text and textual theory became one of the main streams feeding the later critical-text tradition. Even when later editors disagreed with individual readings, the broader shift away from the received text and toward a reconstructed critical text remained in place.
Nestle-Aland and UBS
The supplied booklet notes that Eberhard Nestle’s earliest editions combined readings from Tischendorf, Westcott and Hort, and Weymouth. That matters because it shows how the Cambridge text moved into the later printed apparatus tradition instead of remaining an isolated 1881 curiosity. Modern UBS and Nestle-Aland editions are not identical to Westcott and Hort in every place, but they are historically downstream from that shift.
Why the Bible differences debate persists
For readers committed to the traditional text, this history matters because modern translations frequently inherit a critical-text environment shaped in part by Westcott and Hort’s departure from the received text. For defenders of the critical text, the same history matters because it marks the point at which textual criticism was publicly normalized. Either way, the influence is real enough that the issue cannot be brushed aside as a fringe concern.