We can ask why they might do this. It's likely to be at least partly because they're surrounded by other people speaking English with a French or Dutch accent. This is especially true for McLaren, living in a country where everyone speaks English, but also for Barton, because in the football world there are so many players from all over the world I imagine English is something of a lingua franca in the dressing room. They're bound to be a little bit affected by this and pick up something of the accent.
Another factor is probably that they both have fairly strong regional accents (Merseyside for Barton, and something that sounds like Yorkshire for McLaren), and they will have made an effort to speak more slowly and clearly to non-native speakers. They are therefore not speaking in a completely natural way, and might pick up different mannerisms accordingly.
You know what else we might ask? Are they really speaking in French/Dutch accents at all, that's what. There's a condition called 'foreign accent syndrome', usually a result of brain damage, which causes a person to speak in what sounds like a foreign accent. Family members may be convinced that the sufferer has a New York, Eastern European or Chinese accent, and this can cause problems: one woman in Norway had great difficulties when she began speaking in a German accent in the 1940s. However, it's not actually a specific foreign accent, but simply difficulty in speaking. The accent is never one that listeners are very familiar with, and this is crucial. The brain damage can cause particular problems with speech, which the listener hears as some or other accent. Wikipedia suggests that an American who normally has a rhotic accent (pronounces the 'r' sounds in car park) might have difficulty pronouncing 'r' and therefore omit it. This might sound like a Boston accent to a listener because non-rhoticity is a very salient feature of a stereotypical Boston accent. So is there some other feature of their speech that we're hearing as a foreign accent?
I've had a listen to both, and what follows is entirely unscientific speculation and waffle. You can very definitely hear their original accents in both cases. What is noticeable to me is that they both use very few contractions, saying we are instead of we're, for instance. They also speak slowly, perhaps as a result of the afore-mentioned concession to non-native speakers or possibly even interpreters, in the case of Barton's press conference. I think that some of the filler sounds Barton uses do seem a bit French, but that's because the vowel sounds used in er and similar noises are different from the standard southern English er, and it might well be the same as is used in Merseyside er. Likewise for some of the vowels in his words. His intonation and prosody sound quite Frenchified too: his fairly constant stress placement, for instance, differentiates French from English quite markedly. But it could also be a result of slower speech than he's used to. The frequent final rising intonation he uses is a feature of a Merseyside accent but might well sound French if one was listening in the expectation of hearing one.
I'm just not convinced there's much in it. I think that a combination of some minor changes in their accent and a bit of pareidolia on the listeners' part has made this into something more than it is. One case of foreign accent syndrome involved a woman who went from Geordie to Jamaican, French Canadian, Italian or Slovak. Those are not similar to each other, and there is known (among Geordies, at least) to be some similarity between Geordie and Jamaican. As further evidence, this video is of McLaren supposedly speaking with a Dutch accent, and many of the commenters hear only unremarkable 'northern English'. You make your own mind up.
Wikipedia has this to say on Barton's case, though probably not for much longer:
Barton is now currently under examination by medical professionals and although they are uncertain, the cause of Barton's condition is believed to have been caused when he used both Heads and Shoulders and Aussie Hair Shine Shampoo at the same time. However, this is only an early prognosis.