You are right that this is most likely a Royal Mail issue. I've had several SIMs from Smarty in the past and almost all of them have arrived the next working day.
While you have paid for your first month upfront, the month's service won't start until you activate the SIM, so you won't be losing any of the service you've paid for.
If you're tired of waiting for a SIM to arrive, you might want to just pick one up from a shop. Note that you would need to log out of the Smarty account you've already created before activating the shop-bought SIM, and would need to create a new account (with a different email address) as part of the activation process. Be careful when doing this (do not choose the option to log into an existing account) because lots of people seem to somehow end up ordering a new SIM through the post instead of activating the one they've got from the shop.
Once the shop-bought SIM is up and running, you would then need to log into your original Smarty account and chat to support about getting a refund for your original order.
