The practice you are describing is called peine forte et dure - a little used element in medieval and early modern English law. Peine forte et dure or "pressing" was not intended as a manner of execution. It was a type of legal torture (in a broader sense of the term) used against those who refused to enter a plea after being charged with a crime. The practice dated back to at least the thirteenth century, and the latest known use of the practice was in Cambridgeshire in 1741 (Baker, Introduction to English Legal History 4th ed, pp. 508-509). Perhaps the most famous case of "pressing" was not in England, but in Salem, Massachusetts in 1692 when Giles Corey refused to enter a plea after being charged with witchcraft. Those convicted of witchcraft had their property forfeited, so the story goes that Corey refused to enter a plea, died a Christian & his children were allowed to inherit his estate (Norton, In the Devil's Snare, pp. 277-278; Langbein, Torture and the Law of Proof, pp. 74-77).

As to it's use in the proceedings of the Gunpowder plot, I can find no evidence that the procedure was used by anyone charged, interrogated or convicted in the proceedings. The main secondary work on the Gunpowder plot's proceedings is Nicholl's Investigating Gunpowder Plot, and he makes no mention of the use of "pressing". Further, the only person to be tortured during the proceedings was Guy Fawkes himself, and it's not entirely clear how he was tortured. The warrant for his torture signed by King James stated that "the gentler tortures are to be first used unto him and thus by degrees advancing to the worst", and no further specifics are mentioned. Likely, Fawkes was suspended from his arms and weighted on his legs, but this is conjecture at best. Neither Fawkes nor any of the other interrogated persons were "pressed", for the simple reason that interrogation and punishment were not what "pressing" was for - "pressing" only occurred after a charge had been brought to trial, and the "defendant" (anachronistically) stood silent and refused to enter a plea.

All this is not to say that Catholics were treated harshly during this time period. Perhaps what the writers were drawing on were experiences of Catholics under Elizabeth. A Jesuit priest, Father John Gerard, was arrested in 1594 under suspicion of espionage and attempting to subvert the state on behalf of Catholics, and especially Spain. He was transferred to the Tower of London in 1597, and tortured for six months to extract information about his contacts and doings in England, though largely to no avail. He escaped custody October, 1597 and lived largely on the run, staying with recusants or secret Catholics until 1606. Many of his contacts were arrested during the investigation into the Gunpowder plot, and Gerard was forced to flee the country. He fled to the Netherlands, and in the following years wrote both an autobiography of his time and torture in England, as well as a piece on the state of Catholics in England, using the example of his torture to criticize England more roundly. Gerard's works and his experiences are famous in the historiography, and perhaps inspired the inclusion of a form of torture in the show/movie you mention. It is worth noting that the torture Gerard and others underwent, was only partially due to their religion. In Langbein's list of people legally tortured in the sixteenth and seventeenth centuries, a limited number suffered because of their religion. The majority were tortured to extract information about treasonous acts (suspected and otherwise) or about crimes they had committed (such as murder, burglary, robbery, etc.). And none of the people tortured under these conditions suffered peine forte et dure as a means of execution or punishment.

One Catholic of this era who was subjected to peine forte et dure was Margaret Clitherow of York in 1586. This was for the same reason as Giles Corey in Salem - she refused to enter a plea. The Oxford Dictionary of Saints asserts that "[h]er purpose was to avoid the necessity of friends, servants, and her own children giving evidence against her." This is of course a tertiary source, and I don't know if that motive is attested in the primary sources or was an inference by later writers. (That is also true for Giles Corey - that he was concerned about inheritance is a logical inference, but not explicitly confirmed by the primary record.)