I can see that the Cambridge Dictionary is at least aware of the use of tackle meaning "come to grips with a problem" and I can see that the Sunday Times has used it on occasion. It still seems so connected to (American) football that it shouldn’t be the most idiomatic way for a Brit to express the idea.
Is there any better or more common way to express this idea in British English? or do Brits just use the exact same phrasing but whilst imagining rugby or (field) hockey instead?
As @tchrist pointed out, "address a problem" is more formal than tackle. And in fact, the Americans seem to use it slightly more commonly than the Brits, at least according to this NGram. So I wouldn’t say this is a particularly British expression.
However, my instinct was that actually the most British way to express this was tackle itself. I tried a wild card NGram and if you look carefully (BrE hits are blue and AmE hits are red), you will see that tackle the problem is the most common expression with this sense:
The most common verbs are be, solve and resolve, so tackle comes comes in fourth. And as you can see here Americans also use it, but less. The next verb used with a meaning close to the one you are looking for is also face the problem, but it is "less physical" in connotation.
So the conclusion is that to tackle a problem is actually the most idiomatic phrase used in the UK for that meaning.
Now, it is not clear from your question if you are looking for a synonym phrase that will have the meaning of dealing with the problem but keep the somewhat dynamic connotation of tackle, yet without the football allusion,1 like the example you gave, come to grips with a problem. If that is not the case, what follows may be irrelevant.
The verb grapple is quite physical but has no allusion to any sport. It means:
- to come to grips with (one or more persons), especially to struggle in hand-to-hand combat
- (intransitive; foll by with)
- to cope or contend to grapple with a financial problem2
Another definition I found is:
(grapple with something) to try hard to understand a difficult idea or to solve a difficult problem
- The government continued; e.g. to grapple with the issue of public transport.3
The expression to grapple with a problem is used on both sides of the ocean, though the Brits do have "the upper hand" on it. However, it doesn’t come anywhere near tackle as regards common use. Plus, grapple has this extra connotation that you are dealing with something difficult, sometimes annoyingly difficult, whereas tackle is more neutral towards the problem. It just expresses the dynamic energy with which you deal with a problem, whether difficult or not.
1 I really think that Brits don’t think about football every time they say We need to tackle this problem.