Not being a Brit, my understanding of 'roll on' is based mainly on reading, television, movies, etc. My favorite example is from Raymond Briggs' book Father Christmas. The title character is sitting on a snowy rooftop having a lunch break from delivering presents and listening to a terrible weather forecast on his portable radio - more snow, wind, blizzards, hail - everything he dreads. His comment as he eats his sandwich: "Roll on summer!"
The meaning I've gleaned, from this and other examples, is that summer (or whatever is referenced) is something that's inevitable or regularly scheduled, but the speaker is urging it on - "Roll on!" It can't come soon enough in their opinion, if for no other reason than to get the current situation over with.
In American English, there's probably no commonly used expression that combines the brevity and pithiness with the slightly ironic, sarcastic or rueful tone for urging an early arrival that 'roll on' demonstrates.
There are plenty of all-purpose phrases that urge an event to arrive quickly, like "Come on!" or "Move it!" or "Get a move on!" but those are just as likely to be used to urge a racer or other sports figure to accelerate and win, or to encourage or badger a slowpoke.
Something like "Summer! Can't wait!" is aimed at expressing cheerful eagerness for a positive event, not gloomy expectation that the present situation has gone on far too long.
The best I can come up with it "Hurry sundown!" While not that common, it implies that something bad or irritating will end and something better will arrive, and each day that goes by brings it closer.
Just another example of the UK's way with words.