The following code produces the result I was seeking above:
#Edited code based on Ben Bolker comment
for (j in 1:100) {
for (i in 1:6){
if (runif(1,min=0, max=1) < Prob[i]){
Lattice[i] <- 1}
else{Lattice[i:6] <- 0}
if (Lattice[i] == 0) break()}
resultList[[j]] <- Lattice
dm <- do.call("rbind",resultList) # added in edit
TimeSince <- (apply(dm,2,function(x) which(rev(x)==1)[1])) - 1 # added in edit
}