Oh dear. Bad news. Two of the bullfinches in my garden are sick. They have been sitting around on or near the feeder, fluffed up, lethargic, with their heads held back and with food stuck on their beaks. The symptoms suggest trichomoniasis, a protozoan infection with Trichomonas gallinae. I have only seen it here once before, in a greenfinch a year ago. Other possibilities are infection with Salmonella or Escherichia albertii.
I have reported it to Garden Wildlife Health in which the BTO is a partner. Their usual advice is to keep the feeders clean (which I do anyway) and to suspend feeding if there are several victims. The disease has been endemic in British finches for a decade and caused major mortality in greenfinches in 2006/7. You can download the BTO fact sheet (they call it trichomonosis) here.