Post Pagination Customization (wp_link_pages) Editing Navigation

It makes all pages show “NEXT PAGE”, because you don’t tell it that it should be any different.

What you need is to set that link conditionally based on which page is currently displayed:

wp_link_pages( array(
    'before' => '<div id="slideshow">',
    'after' => '</div>',
    'next_or_number' => 'next', 
    'previouspagelink' => '<span id="previous" style="display:none;"> PREVIOUS SLIDE </span>',  
    'nextpagelink' => ( get_query_var('page') < 2 ) ? '<span id="next"> WELCOME </span>' : '<span id="next"> NEXT PAGE </span>'
) );

This uses a conditional statement:

( get_query_var('page') < 2 ) ? '<span id="next"> WELCOME </span>' : '<span id="next"> NEXT PAGE </span>'

which checks the current page (you can get it by checking page query var) and returning one of the strings based on the condition (here we check if it’s the first page or any other).

You can read more on page query var here: What is the difference between $paged and $page?