Hi Dylan.  Yes you should be able to get the old black and white photos printed in store. Naturally they would have to be put onto a CD from your computer to get them there or if you have a card reader/writer you could transport them on one of your camera memory card. Remember to save them as .jpg files.
Preparation makes sense and it helps if you use your graphics package to crop the photo to the correct aspect ratio.
 This prevents the automated cropping of your image in store, so That its you who decides what is cropped away, if any,, rather than tha ASDA computer if you see what i mean A lot of the old b and w photos used to be a different aspect ratio to modern ones.Â
 Another way to maintain shape integrity would be to create a new background in any colour other than white. This new background should be in the correct aspect ratio as ASDAs capability eg 6x4, 7x5, 9x6 etc   Then scan your photo and place it onto the background.
If you wish you could make the background into a printed frame in any design to suit you project
Smaller photos could be put 2 or more into the same background area side by side if you don't particularly want enlargements.
This will create a copy of your photo with some form of border, which after ASDA processing can be cut off with scissors if appropriate.
This is one of the techniques my son adopts at the
www.lazypics.com photo studio
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Andy, the fotofool