My answer is just based on limited experience (so you might defer to others for more than anecdotes).
In my semi-arid climate, in 2018, I grew tomatoes with only a path (maybe three feet wide) separating it from our thornless blackberries (which blackberries had been there for many years). The tomatoes did not contract any diseases from the blackberries, nor vice versa. They're such different plants in their growth habits that I wouldn't expect people would normally plant them close enough to each other for it to matter. It's possible that it would be a bigger issue in a humid climate where fungal diseases are more of a problem. Our blackberries and tomatoes were just directly in the ground.
In 2019, I had a few tomatoes at about the same distance from the blackberries, with no issues. One of the tomatoes actually was diseased, probably with some kind of Alternaria (but the disease came with it from the store, and didn't seem to bother the blackberries, although it did spread to some other tomatoes a fair distance away). The tomato wasn't bothered terribly by the disease (but I could tell it had one).
In 2020, this year, I have tomatoes probably about 15 feet from the blackberries. They all seem healthy enough. Ditto for 2016, and 2017.
Now strawberries, on the other hand, I planted a tomato (Tidy Rose F1) in the middle of a strawberry patch (the ground was mulched, too), and it got severely diseased and died (while the tomatoes further away, which were not with strawberries, did not get that disease).
We had potatoes on the other side of the path from the blackberries, this year. The potatoes did not do well (they looked diseased to me, although we harvested and ate them), but the blackberries were fine.
Anyway, it might be wise to be careful and not put them super close, but I wouldn't bother extending lots of effort putting them zillions of feet away.
I had never heard before about keeping nightshades and blackberries separate. I had heard about keeping blackberries and raspberries separate, though.