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#C, 138 135 bytes
char*s="aiueokstnhmyrw";i,j;k(*v){j=strchr(s,*v>>8)-s;i=strchr(s,*v)-s;j=i<4?i++:j<4?j-~(2520%i&3):!++i;printf("%c%c",s[i],i<5?:s[j]);}
Wandbox
#C, 139138 138135 bytes
char*s="aiueokstnhmyrw";i,j;k(char*v*v){for(;*v;i||j=strchr(i=j)s,*v>>8)j=strchr-s;i=strchr(s,*v++*v)-s;j=i<4?i++:j<4?j-~(2520%i&3):!++i;printf("%c%c",s[i],i<5?:s[j]);}
WandboxWandbox
#C, 139 138 bytes
char*s="aiueokstnhmyrw";i,j;k(char*v){for(;*v;i||(i=j))j=strchr(s,*v++)-s;j=i<4?i++:j<4?j-~(2520%i&3):!++i;printf("%c%c",s[i],i<5?:s[j]);}
#C, 141139 139138 bytes
char*s="aiueokstnhmyrw";i,j;k(char*v){for(;*v;i||(i=j))j=strchr(s,*v++)-s;j=i<4?i++:j<4?j+1+j-~(2520%i&3):!++i;printf("%c%c",s[i],i<5?:s[j]);}
#C, 141 139 bytes
char*s="aiueokstnhmyrw";i,j;k(char*v){for(;*v;i||(i=j))j=strchr(s,*v++)-s;j=i<4?i++:j<4?j+1+(2520%i&3):!++i;printf("%c%c",s[i],i<5?:s[j]);}
