I've been watching it on my TV since it first launched in the US. They have a Roku app. Yes, buying a Roku is another expense, but they are available for $30 USD (23 GBP). This page from Roku lists places to buy it in the UK.
Sherlock is not on britbox, in the US. It is in the UK, but not the US.
Unless you mean the series from the 80's, which is on amazon: https://www.amazon.com/A-Scandal-In-Bohemia/dp/B074PD6FMW