How do I insert a tie (~, non-breaking space, tilde) under expl3 syntax?

This question has two distinct parts, one of which is easier than the other!

The specific issue here is (presumably) not inserting a tilde but rather inserting a non-breaking space in some 'fixed' text stores as within a LaTeX3 tl variable. While a tilde is a convenient shortcut for this at the document level, here I think I would simply use the macro nobreakspace.

tl_set:Nn l_silly_tl { Text ~ and ~ a nobreakspace tie:?? }

(Note that at present there is no expl3 code-level 'non-breaking space' function.)

On the wider issue of 'inserting non-standard catcode characters', the expandable function char_generate:nn is available to create tokens of (almost) all reasonable category codes

tl_set:Nx l_silly_tl
  {
    char_generate:nn { `a } { 4 }
  }

There are a few cases that are not covered by this approach. For example, with older expl3 releases we couldn't do active characters and you'd need to use the regex engine:

RequirePackage{expl3}
ExplSyntaxOn
tl_clear:N l_tmpa_tl
regex_replace_all:nnN { } { cA~ } l_tmpa_tl 

More recent versions of expl3 can do this using char_generate:nn, so it's more of a historical note.

The second option is stick to the 'traditional' lowercase trick, using expl3 naming of course

group_begin:
char_set_catcode_active:n { `@ }
char_set_lccode:nn { `@ } { `~ }
tex_lowercase:D
   {
     group_end:
     tl_set:Nn l_silly_tl { Text ~ and ~ a @ tie:?? }
   }

(Notice that there is no public interface to lowercase beyond the saved-primitive.)