App scripts registered by Shopify apps' are loaded once the page is loaded successfully and it causes delay in loading of cart drawer. You can add the following line of code in your theme.liquid file before ending </head> tag and it will start opening the cart drawer all the times:
<!-- Upsell Cart Drawer -->
<script type="text/javascript"
async="async"
src="https://cdn.shopify.com/s/files/1/0537/1265/0427/t/3/assets/upsell-cart-drawer___shopify.js?shop={{shop.permanent_domain}}&t={{ 'now' | date: "%Y%m%d-%H"}}-01"></script>
<!-- /Upsell Cart Drawer -->