What is the ruling on participating in a company, affiliated with a British bank, that buys and sells currencies on the global online exchange, whether the participant does so himself or it is done automatically by the company's software?

We have no knowledge of the circulating system. However, trading in currencies is more severe than other forms of trade and is fraught with Shar'i prohibitions. If the Shar'i controls are met, there is no objection to practicing it through international stock exchanges, provided that the Shar'i controls are confirmed to be present and the prohibitions are absent.