I haven't seen the movie in quite a while, but I can confirm that the shots in the beginning (when Sidney Poitier is being landed on board the ship from a helicopter) show a British ship. It's clearly visible for only a few seconds, but I think it's a frigate or something of that sort - smaller than a destroyer.
The movie was made by a British company; I imagine most of the "on board" footage was shot on a sound stage. I think some stock footage of American and/or British ships may also have crept in. The distant shots of the Bedford pretty clearly were of a model. My vague recollection is that it looked like a Charles F. Adams class destroyer, but I could well be mistaken about that.
I liked the movie - and I liked the book better. I remember reading it in high school (which shows once again what an Olde Phogey I am) and being proud of myself for figuring out that it was a modernized retelling of Moby Dick. (Looking back on it now, a reader would have to be downright obtuse to miss that. The Sidney Poitier character survives the sinking of the Bedford, drifts around for a while on a raft, and eventually gets picked up by the Russian trawler, "which found only another orphan.") As I recall, the movie makers changed the ending in several other significant ways; the former U-boat captain who was on board as a NATO advisor played a considerably bigger role in the book. I imagine it's been out of print for decades, but it would be worth seeking out.
Youth, talent, hard work, and enthusiasm are no match for old age and treachery.