One approach is to check the place of manufacture. In the US market, film
made in Germany is Agfa (but that's going away fast, since Agfa has
folded), film made in Italy is Ferrania, and film made in Japan is
Konica-Minolta or possibly Fuji. Details may vary in other countries,
though; several manufacturers have facilities in multiple countries, so
what gets supplied to one market may originate somewhere other than the
film that's sold to another market.

Another method is to check the plastic canisters in which the film
cartridges are shipped. Each manufacturer has a distinctive canister
design. These can be harder to describe, but here goes: Agfa and
Konica-Minolta canisters have caps that fit around the outside rim of the
canister; Agfa's are black whereas the Konica-Minolta canisters are
translucent. Fuji and Ferrania canisters have caps that fit inside the rim
of the canister. Both are usually translucent, although I've seen some
black Ferrania canisters. Kodak's canisters have caps that fit outside the
rim, like the Agfa and Konica-Minolta canisters, but the caps are
flat-topped. (This is largely moot, though, since Kodak doesn't permit its
film to be sold under other names.)

A third method is to examine the finished negatives. I don't have samples
handy, but there are distinct differences, even aside from bar codes.
Frame number marking style and location, dots or lines, etc., are all
unique. Identifying the film by these markings isn't hard once you've seen
enough samples of each type. Recent Ferrania film may be easiest; they've
taken to marking their film "Ferrania," although this wasn't true a few
years ago.

All of the above applies to color film. The B&W market has different
players. Ilford used to supply a lot of house brands, but they've
reportedly stopped doing so. I don't know about the UK market, but in the
US, there are precious few house-brand B&W films; the only ones I'm aware
of are sold by Freestyle (a California mail-order outfit). They sell Foma
(Czech) and Forte (Hungary, IIRC) film. They used to sell Ilford and Agfa
film, too, but those supplies are dwindling at this point. I've never used
Forte film, so I can't help with further identifying characteristics for
it. The Foma film I've bought has come with caps that fit inside the film
canister. Some of these canisters have been black and others have been
translucent with blue caps.