As I said you have to use dates instead of characters, see this example
library(dplyr)
mydata<- read.table(text="ID Lastupdate
A '2020-04-29 08:02:00'
B '2020-05-20 09:04:12'
C '2020-05-18 10:23:14'
D '2020-05-06 12:00:45'
E '2020-05-10 13:21:45' ", header=T)
mydata %>%
mutate(Lastupdate = as.POSIXct(Lastupdate),
updatecondition = ifelse(Lastupdate > as.POSIXct("2020-05-10 08:00:00"), 'Yes', 'No'))
#> ID Lastupdate updatecondition
#> 1 A 2020-04-29 08:02:00 No
#> 2 B 2020-05-20 09:04:12 Yes
#> 3 C 2020-05-18 10:23:14 Yes
#> 4 D 2020-05-06 12:00:45 No
#> 5 E 2020-05-10 13:21:45 Yes
Created on 2020-05-29 by the reprex package (v0.3.0)