Is there a more British way to talk about tackling problems?

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:

1. to come to grips with (one or more persons), especially to struggle in hand-to-hand combat
2. (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.}

^2Collins

^3Macmillan

The essence of the English language (and many others for that matter) is that words expand their meanings by being applied figuratively to different situations. The native speaker uses the context of the word to interpret its meaning without confusion.

In the case of “tackle” no British person would be confused by the fact that it is an everyday word to deal with a problem, and an everyday word to describe an action in several team games (although American football would be unlikely to come to mind any more than Aussie-rules football).

The ngram of @fev confirms my feeling that tackle is most commonly used word in everyday speech, and alternatives like “address” are regarded as academic or unnatural, as several have commented.

I find it difficult to follow the logic of the statement that:

“It still seems so connected to (American) football that it shouldn't be the most idiomatic way for a Brit to express the idea.”

Why should a Briton be aware of or be concerned about a mental association of the word in different culture, even assuming awareness of it? No more than a German should (or would) be concerned about the copulatory associations an English speaker might have with the German word for ‘fox’ or the bowel sounds the former might associate with exit signs on the Autobahn.

Of course, it is helpful if the individual travelling between countries is aware of the situation. There is no problem in Britain if a student says to another student “Lend me your rubber.” However if he spends a year in the US he needs to learn to say “Lend me your eraser.” Two countries divided by a common language.