We present 15-GHz follow-up radio observations of 11 Swift gamma-ray burst (GRB) sources, obtained with the Arcminute Microkelvin Imager Large Array (AMI-LA). The initial follow-up observation for each source was made in a fully automated fashion; as a result four observations were initiated within 5 min of the GRB alert time stamp. These observations provide the first millijansky-level constraints on prolonged radio emission from GRBs within the first hour post-burst. While no radio emission within the first six hours after the GRB is detected in this preliminary analysis, radio afterglow is detected from one of the GRBs (GRB 120326A) on a time-scale of days. The observations were made as part of an ongoing programme to use AMI-LA as a systematic follow-up tool for transients at radio frequencies. In addition to the preliminary results, we explain how we have created an easily extensible automated follow-up system, describing new software tools developed for astronomical transient alert distribution, automatic requesting of target-of-opportunity observations and robotic control of the observatory.