Jersey doesn't require British citizens to present a passport:
http://jersey.com/English/transport/Pages/defa ult.aspx
The only scheduled flights on that route are with FlyBe. Like all airlines, they require passengers to present ID for travel on domestic flights but they accept all of the following. (Directly cut & pasted from FlyBe's website):
* A valid passport
* An expired passport (can be used on domestic flights for up to two
years after expiry)
* Valid photographic EU or Swiss national identity card
* Valid photographic driving licence full/provisional
* Valid armed forces identity card
* Valid police warrant card/badge
* Valid airport employees security identity pass
* A child on parent' s passport is an acceptable form of ID
* CitizenCard
* Valid photographic firearm certificate
* Valid Government-issued identity card
* Electoral identity card
* NUS cards photographic (National Union of Students)
* Photographic University/College ID card
* Company ID cards of Nationally recognised companies (photographic)
* Council issued bus pass (Senior Citizens only)
* Pension book or Post Office card (as only acceptable forms of non-photographic
identification)
I'm confident that the airline's 'domestic' rules will apply to flights to the Channel Islands (even though they're not part of the UK) but a quick call to 0871 700 2000 should enable you to get confirmation. (10p per minute, 0800-1830 Mon-Fri, 0800-1600 Sat, 1000-1800 Sun).
Chris