The question is still not clear, please rewrite it like:

1. I do X.
2. I expect Y.
3. But I get Z.

• The question's current code seems to work as expected on both browsers (Using Tampermonkey on Chrome).

• A setTimeout or AJAX tricks don't seem to be needed to wait for, or access, that object.

• If you mean that you can't use start_up from the console, that's the nature of the particular with_jquery() function being used. It wraps your function in parentheses () which effectively makes it anonymous.

I recommend never doing jQuery that way from a userscript. 99% of the time, use @require and a @grant setting other than none.

On friendly, jQuery powered pages, where heavy interaction with page objects is desired (like Stack Exchange sites), use the page's jQuery like this:

// ==UserScript==
// @name        (SE) Inspect global object
// @match       *://*.stackoverflow.com/*
// @match       *://*.stackapps.com/*
// @grant none
// ==/UserScript==

//var start_up = function( $ ) {    // BAD
function start_up ( $ ) {           // GOOD
    console.log (
        "===> ",
        typeof StackExchange.options.routeName,
        StackExchange.options.routeName
    );
};

withPages_jQuery (start_up);

function withPages_jQuery (NAMED_FunctionToRun) {
    //--- Use named functions for clarity and debugging...
    var funcText        = NAMED_FunctionToRun.toString ();
    var funcName        = funcText.replace (/^function\s+(\w+)\s*\((.|\n|\r)+$/, "$1");
    var script          = document.createElement ("script");
    script.textContent  = funcText + "\n\n";
    script.textContent += 'jQuery(document).ready( function () {' + funcName + '(jQuery);} );';
    document.body.appendChild (script);
};

This allows you to see your function (start_up) and use or debug it from the console.