A lot older films than that were originally shot on film in full HD
That's not saying what's broadcast on E4 HD isn't converted or upscaled though.
Film isn't shot in 'HD' as such, HD is a digital domestic video TV standard.
Films have been routinely shot using 35mm film stock for years. This has resolution comparable with modern Full HD TV, so any old films which have been recently transferred to video should look great in HD
More recently, a lot of films have been shot on 70mm film stock (and shown in IMAX cinemas). This equates nicely with the new, incoming format of 4K.
Once it's in the digital domain, you can make it look however you like: B&W, Technicolor, DeLuxe colour, etc, etc. And you can play around with the 'grain' and all sorts. Just like Photoshop, but with moving pictures.
And of course these days, most feature films are shot on digital video cameras and recorded straight to Hard Drives anyway! So there can actually be no 'film' at all between when the movie was shot, and shown at your local cinema!