Bradshaw’s father, the Duke of Grafton’s factotum, left his affairs in disarray and his family became state pensioners, except for Robert, who inherited the reversion of the lucrative auditorship of plantations and was meanwhile placed in the Pay Office. By 1800 he was the canal-building 3rd Duke of Bridgwater’s man of business, in much the same way as his father had attached himself to Grafton. By Bridgwater’s will in 1803 he was appointed one of the three trustees of the estate and sole superintendent at a salary of £2,000 p.a. plus expenses, with power to name his successor. He was provided with residences at Worsley and Runcorn. In 1810 he doubled canal tolls, the average profit from 1806 to 1826 being £45,000 p.a. (about 13 per cent). He also prospered on his own account, investing in the sale of coal in Lancashire, in East India stock and in a country estate at Woodmans, Hertfordshire. When he was manoeuvred into resignation of his position in 1834, he contracted to sell his properties near the Bridgwater canal for £47,000, as well as another estate near Brackley.
Shortly before the duke died, he returned Bradshaw for his pocket borough of Brackley, intending that he should be attentive to his canal interest and shield its monopoly. Bradshaw retained the seat and a say in the choice of his colleague (under the Marquess of Stafford’s patronage) for 30 years, unopposed. He is not known to have spoken in debate before 1820 and made little mark at Westminster, where he followed his patron’s line intermittently. He voted for Pitt’s question for the orders of the day, 3 June 1803, and his motion for naval inquiry, 15 Mar. 1804. He was listed a supporter of Pitt then and during his second administration.
He went on to support Grenville’s ministry, voting for Brand’s motion following their dismissal, 9 Apr. 1807. In February he had asked the prime minister for promotion for his son James in the navy.
Bradshaw died 8 Jan. 1835, ‘beyond all doubt’, his precondition for the fastening down of his coffin.