It could be less about the period and more about the specific social/cultural context.
According to this BBC interview with sports historian Tony Collins (who has written multiple books on the history of rugby), professional rugby player Tony Marchant, and Trever Gibbons (rugby league writer):
TONY COLLINS: I mean the other thing that I, I found amazing when I went there a couple of years ago, in Australia, is the fact that rugby league, well certainly in New South Wales and s- and Queensland, rugby league is football, so if you talk about rugby, they think you're talking about rugby union.
TREVOR: But, but, quite often in our house, especially growing up, you wouldn't talk about rugby - we would say, "Are we off to the football today?" And we'd know we meant rugby. You wouldn't say, "Are you off to the rugby?" We'd have to say, "Are you off to the football?" And like if somebody's having a- or if a team's having a good game we still say they're playing some great football.
TONY COLLINS: Cos I would have, I would have thought that, that anybody, we say rugger it's obviously rugby union but, yeah, I mean my granny used to call rugby league, football. [...] Yeah, well it's like Hull Football Club. Because that's what football was.
If the Walkers came from a "rugby town" - or rather, given their evident social class, if John's public school was a "rugby school" - then it might be normal for them to say "football" meaning rugby. Note that fifteen players means it's rugby union rather than the thirteen-player rugby league, which makes sense (and makes the above quoted interview less directly relevant) since rugby union is the posh version of the sport, and the Walker family are obviously a posh family rather than working-class northerners. We don't know exactly which school any of them went to, although Jim Brading went to Rugby School, like Arthur Ransome himself - this would be the school that the author would be most familiar with. Presumably rugby is one of the most important sports there, and surely there of all places it might be normal to use simply the word "football" to refer to rugby union football, rather than "rugby" which is the name of the school itself!