Ageratum? Though that tends to be more lilacy than blue. Otherwise blue lobelia, or blue pansies. Echium 'blue bedder' is good, but gets 9-12 inches tall.

Hello and welcome to Goy
I had beautiful trailing blue lobelia last year which lasted to November Geoffreyneil

Anagallis is a really blue blue! :-)

I'd do it in red, white and blue petunias

I'm with Spritz on the Anagalis - it is spectacular in full sun!
Scaevola is good in baskets, as is Torenia. I've wondered about Linum and hardy Geranium in baskets, as they can get leggy and would tumble.

Problem is that the Begonias are large blooms compared to all the suggestions (except Petunia, which on a personal level I hate (sorry Andy!)), and the good Pansies in blue tend to be sky blue.

Do the other baskets have to be Begonia?

T&M are doing a red white and blue petunia collection.
http://www.thompson-morgan.com/flowers/flower-plants/petunia-plants/petunia-easy-wave-union-jack-mix/p94657TM

I love Spritz's suggestion... But I would though

I would go for trailing blue Lobelia too..almost maintenance free,just a haircut now and then..

I'd query just how blue the blue petunia is though - all the ones I've seen have been purplish rather than a proper blue. There is an upright version of dark blue flowered Lobelia too, if you didn't want the whole blue basket to be sprawling.
Re the size of flowers comment above - if you're using bedding begonias, the flowers won't be too large in comparison, it's only the trailing corm (tuberhybrida) ones that have large flowers.