alias server_uptime='ssh $host 'uptime -p''

alias server_uptime='ssh $host '"'uptime -p'"

In the first case, the user has four single quotes on a line, wishfully hoping that the shell will match them up as outer quotes around a string with literal single quotes:

#                   v--------match--------v
alias server_uptime='ssh $host 'uptime -p''
#                              ^--match--^

The shell, meanwhile, always terminates single quoted strings at the first possible single quote:

#                   v---match--v
alias server_uptime='ssh $host 'uptime -p''
#                                        ^^

Which is the same thing as alias server_uptime='ssh $host uptime' -p.

There is no way to nest single quotes. However, single quotes can be placed literally in double quotes, so we can instead concatenate a single quoted string and a double quoted string:

#                   v--match---v
alias server_uptime='ssh $host '"'uptime -p'"
#                               ^---match---^

This results in an alias with embedded single quotes.

None.