Did you see this on TV? If so do you remember about what year and where you were living? The reason for asking those two questions is that prior to AMC etc. and other cable stations your local broadcast stations would often run old movies as afternoon movies, Sunday matinees, late night movies etc. If this were the case it would be possible to identify the stations that were in your area. Once that was done it would be possible to look up old program listings in the local TV Guides and similar publications.
Now another question or three. Do you recall at all whether the scene was in the beginning, middle or end? Do you remember at all a sense of whether the movie was set here in the US or did it have a mix of countries? Can you remember or do you have a sense it was a long feature, i.e. about 80-90 minutes, or shorter like 50-70 minutes. There are several reasons for these questions. Regarding the placement of the scene it might lead us as to whether the scene was close to the typical "good feeling ending" of the old movies or whether there was more substantial plot after the scene. The reason for the setting is because many of the old movies did mix settings and the British film industry was strong along with Hollywood. So sometimes movies involving musicals, light comedy/romance etc. would have the musicians portrayed as traveling to Europe etc. It sort of added to the glamour, escape aspect etc. for the viewer. One of the most helpful things would be about the length. If it was a shorter duration feature then it helps to narrow down which studios would be the most likely ones to have produced the movie. RKO for example although their were others as well. Many of the B, C and D movies fit the shorter running time. It's possible that your movie is in the realm of B,C or D studios. A lot of that content is gone or is around but not on a DVD, streaming etc. There are literally hundreds of thousands of movies of all kinds and the sorting and selecting of what to spend the money on transferring to DVD leaves some things favored over others.
But there is a caveat to this sleuthing. Some things that used to be available for "reruns" up into the '60's even are no more. There may be remnants of video or little pieces of information here or there. Especially on the movies from smaller operations. The MGM, Fox and Paramount stuff is more intact than most.
Having said this the most likely avenue to research this is the aspect of trying to zero in more closely on when you saw the movie and where you lived. The old TV Guides etc. usually gave a decent short summation of the plot, main actors and what year the movie was from. Since they were weekly you would have a few hours going through a few years of archived copies at your local big city library and making notes or if you could find them archived online.
A couple of last ditch ways would maybe yield an answer also. The nostalgia movie sites, movie fan sites etc. sometimes have a "columnist" who answers submitted questions like yours. Getting them to respond to any one particular viewer question is a real shot in the dark because you can imagine how many thousands of questions they get every week. One other is there are nostalgia movie/TV companies that produce DVD's of old TV series, old movies, film collections of certain topics, stars, directors etc. I used to get buy some of them and they would send out a "monthly" 12 page or so "catalogue" of their offerings including new ones just completed. Many of those were of the sort of movie you're talking about where it wasn't something famous and well known but it was memorable to some viewers. I had some great stuff from RKO, American International, Du Mont etc. Not expensive and they gave me many hours of enjoyment as I love the old stuff and especially black and white. Here are a couple of sites for that. You can comb through their listings and descriptions of each thing they have and see if anything pans out. Free to browse and if you buy I have found the prices to be reasonable. Richards is really good. I have used all of these and they are all fine. In the case of Brians that obituary section for B-movie stars is very helpful because it gives a listing of some of the movies the star was in that are not commonly known or the listing will give tangential information to keep you going further.
https://classicmoviesandtv.com/channel/movietv
https://richardsoldmovies.com/
https://moviebuffsforever.com/
https://rarefilm.net/
http://www.briansdriveintheater.com/dead.html
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