The Postman.
The Postman is the character who hears Bernard's wish, to avoid being late and to have more time, then he gives him the watch. His full name is never mentioned though his first name is revealed to be Gilbert in the episode Time's Up. He is from the firm which maintains the watches and enforces the rules for keeping them. He drives a van, which can travel through time, with the logo of the firm which has a wing and the motto "Time flies but we stand still". He is summoned when his clients want him to be there and help them with any problems with their watch. However he is shown to be sometimes forgetful such as not collecting the watch when it needed servicing, until Bernard summons him for help, although it was overdue, not mentioning a client getting a new watch when crossing time zones or forgetting his van has a faulty clutch which runs away with the clients' life files aboard when he left the engine running. The only other postman from the firm to be seen in the show is his American colleague, whose name is not revealed, in Time's Up who comes to give Bernard his American watch almost immediately after Gilbert collects the British watch.
The Postman with his American colleague.
The Postman's van in between times.