I like John Christopher. His Prince in Waiting and Lotus Caves were key books for me, growing up, and I've enjoyed many of his other books as well. This book sees him applying his solid Young Adult approach to another SF trope - in this case, alternate worlds.
The problem is that the story feels very formulaic - the same basic tone as Christopher's other YA stories, with a slightly different plot; young, uncertain boy doing his best to stumble toward adulthood in strange circumstances. The protagonist is appealing, the writing is smooth, but there's nothing new here.
Brad, the American cousin, is simply not credible - he speaks Latin, Algonquin, has an eidetic memory, is athletic, knows how to build machines and sail boats, and is charming and diplomatic. The idea is presumably to illustrate protagonist Simon's general mediocrity (and good heart), but Brad is so overdrawn that it just doesn't work.
The plot moves along well enough, until it simply moves too fast. The boys (mostly the talented Brad) change their new world dramatically - all in the space of a few months, it seems. That done, they move on to the next place. It's one thing to gloss over boring details. It's another to race through them with no regard for credibility. It's a shame, since the setting (a version of the Roman Empire) is otherwise thoughtfully detailed. I wish we'd had more chance to see it as it rushed past.
In short, written with Christopher's usual skill, but sketchily plotted, as if his mind were on other things. I don't have much interest in the sequels.