<?php

$live_quotations = [
    'June 2015' => [
        'MEDIUM' => [ 0.00 ],
        'LOW' =>    [ 160.50 ],
        'HIGH' =>   [ 60.80 ]
	],
	
    'July 2015' => [
        'HIGH' =>   [ 121.96 ],
        'MEDIUM' => [ 263.00 ],
        'LOW' =>    [ 226.00 ]
	]
];

function cmp($a, $b) {
    $order = [ 'LOW' => 0, 'MEDIUM' => 1, 'HIGH' => 2 ];
    return $order[$a] - $order[$b];
}

foreach($live_quotations as $exp_conversion_date => $Aconversion_likelihood){
    foreach($Aconversion_likelihood as $conversion_likelihood => $quotation_values){

        uksort($live_quotations[$exp_conversion_date], "cmp");

    }
}

var_dump($live_quotations);