#C, 153 bytes
char*s="aiueokstnhmyrw";i,j;k(char*v){i=strchr(s,*v++)-s;*v&&(j=strchr(s,*v)-s);i<5?i++:j<4?j+=(i>10&&i%2?i%9:1):(j=!++i);printf("%c%c",s[i],i<5?:s[j]);}
#C, 153 bytes
char*s="aiueokstnhmyrw";i,j;k(char*v){i=strchr(s,*v++)-s;*v&&(j=strchr(s,*v)-s);i<5?i++:j<4?j+=(i>10&&i%2?i%9:1):(j=!++i);printf("%c%c",s[i],i<5?:s[j]);}