Bradbury, who turned professional in July and only received an invite to the first tournament of the new DP World Tour season on Friday, added a 66 to his opening 63 at Houghton Golf Club to set the clubhouse target on 13 under par.
South Africa's Casey Jarvis and France's Romain Langasque had reached 12 under after 14 and 11 holes respectively before play was called off for the day following an earlier stoppage of almost three hours due to lightning.
Bradbury, who is ranked 1397th in the world, recovered from a bogey on the second with an eagle and six birdies in the next nine holes and at one point enjoyed a seven-shot lead before dropping shots on the 15th and 16th.
"It's the same as yesterday, I can't complain," the 23-year-old from Wakefield said. "I had it going there on the front nine and thought it was going to be another 63, maybe 62.
"On the back nine I made a few mistakes but overall very, very happy."
Asked about his lofty position in just his fifth tournament in the paid ranks, Bradbury added: "You always dream of it don't you?
"That's what you try and do when you show up but it is a little bit of a surprise considering my preparation. I'm just happy to be where I am."
World number 74 Christiaan Bezuidenhout, the top ranked player in the field, is three shots behind Bradbury following a flawless 64, while defending champion Thriston Lawrence was six under with seven holes of his second round to play.