hyperprob <- function(y, nd, nt, p, lower.tail=T, offset=0) {
  phyper(y, nt*p, nt*(1-p), nd)/2 + #dhyper does not deal with fractional gamma well
    phyper(y-lower.tail, nt*p, nt*(1-p), nd, lower.tail=lower.tail)/2 +
    offset
}

uniroot(hyperprob, c(0,1), y=y, nd=nd, nt=nt, lower.tail=F, offset=-0.5)
uniroot(hyperprob, c(0,1), y=y, nd=nd, nt=nt, lower.tail=F, offset=-0.025)
# estimate and lower are 0 if every sample is 0
upper <- uniroot(hyperprob, c(0,1), y=y, nd=nd, nt=nt, lower.tail=T, offset=-0.025)