Ubercart

You can override the theme_uc_cart_block_title() function in your theme. This is the most general method, and learning how to override theme functions will help you do a lot of customization, not just renaming text. Or, you can enable the Locale module and use that to "translate" all occurrences of "Shopping cart" to whatever you want. This method only customizes the text strings for you, but it too can be useful.

This is something I'm sure a lot of Drupal users outside of the USA would want to be able to do. Is it possible to amend Ubercart core to use a variable all over the module(s) for the name of the cart/basket/trolley (whatever you want to call it) and have a configuration setting for naming the cart?

This way, one change would replicate to all the areas the term is used, such as the buttons, blocks etc. Maybe even assign it a token, so if you change the text on the button it can be "Submit to ['uc:cart']"?

This is something way beyond my capabilities as a non-programmer (also, forgive the above token syntax if it's incorrect), but am I onto something or just talking gibberish?

There's a module for this! It's called Ubercart Basket.