Here are 2 ways you could do this.
Using dplyr package
You could use case_when from the dplyr package to do this.
library(dplyr)
country_name <-c("Finland", "Denmark", "Switzerland","Iceland", "Netherlands", "Norway", "Sweden", "Luxembourg", "New Zealand",
                 "Austria", "Australia", "Israel")
nt_final_table <- data.frame(country_name)
first_world_countries <- c("Australia","Austria","Belgium","Canada","Denmark","France","Germany","Greece","Iceland","Ireland","Israel","Italy","Japan","Luxembourg","Netherlands","New Zealand","Norway","Portugal","South Korea", "Spain","Sweden","Switzerland","Turkey","United Kingdom","USA")
second_world_countries <- c("Albania","Armenia","Azerbaijan","Belarus","Bosnia and Herzegovina","Bulgaria","China","Croatia","Cuba","Czech Republic","EastGermany","Estonia","Georgia","Hungary","Kazakhstan","Kyrgyzstan","Laos","Poland","Romania","Russia","Serbia","Slovakia","Slovenia","Tajikistan","Turkmenistan","Ukraine","Uzbekistan","Vietnam")
third_world_countries <- c("Somalia","Niger","South Sudan")
nt_final_table_categorized <- nt_final_table %>% mutate(category = case_when(country_name %in% first_world_countries ~ "First",
                                               country_name %in% second_world_countries ~ "Second",
                                               country_name %in% third_world_countries ~ "Third",
                                               TRUE ~"Not listed"))
nt_final_table_categorized
Sample output
   country_name   category
1       Finland Not listed
2       Denmark      First
3   Switzerland      First
4       Iceland      First
5   Netherlands      First
6        Norway      First
7        Sweden      First
8    Luxembourg      First
9   New Zealand      First
10      Austria      First
11    Australia      First
12       Israel      First
Using base R
In base R we could create a data frame that lists the countries and their category then use merge to perform a left-join on the 2 dataframes.
country_name <-c("Finland", "Denmark", "Switzerland","Iceland", "Netherlands", "Norway", "Sweden", "Luxembourg", "New Zealand",
                 "Austria", "Australia", "Israel")
nt_final_table <- data.frame(country_name)
first_world_countries <- c("Australia","Austria","Belgium","Canada","Denmark","France","Germany","Greece","Iceland","Ireland","Israel","Italy","Japan","Luxembourg","Netherlands","New Zealand","Norway","Portugal","South Korea", "Spain","Sweden","Switzerland","Turkey","United Kingdom","USA")
second_world_countries <- c("Albania","Armenia","Azerbaijan","Belarus","Bosnia and Herzegovina","Bulgaria","China","Croatia","Cuba","Czech Republic","EastGermany","Estonia","Georgia","Hungary","Kazakhstan","Kyrgyzstan","Laos","Poland","Romania","Russia","Serbia","Slovakia","Slovenia","Tajikistan","Turkmenistan","Ukraine","Uzbekistan","Vietnam")
third_world_countries <- c("Somalia","Niger","South Sudan")
country_name <- c(first_world_countries,second_world_countries,third_world_countries)
categories <- c(rep("First", length(first_world_countries)),
                rep("Second",length(second_world_countries)),
                rep("Third",length(third_world_countries)))
all_countries_categorised <- data.frame(country_name, categories)
nt_final_table_categorized <-merge(nt_final_table, all_countries_categorised, by ="country_name", all.x=TRUE)
nt_final_table_categorized
Sample output
   country_name categories
1     Australia      First
2       Austria      First
3       Denmark      First
4       Finland       <NA>
5       Iceland      First
6        Israel      First
7    Luxembourg      First
8   Netherlands      First
9   New Zealand      First
10       Norway      First
11       Sweden      First
12  Switzerland      First