Thomas Cromwell was arrested and eventually executed by Henry VIII, although the reasons for his execution are speculative and probably due to a combination of factors including falling out of favour with the king and competition with his political rivals. One could perhaps assume he was another victim of the kings ruthlessness and paranoia.
Do you think that the treatment and execution of his famous ancestor had any influence on Oliver Cromwells views of the monarchy prior to the issues leading to the English Civil War and whether any existing opinions led to his future decision making during and after the civil war?
This question cannot really have a definitive answer because it asks about the internal motivations and opinions of a person long since dead. What we can do, is look at the items Oliver Cromwell wrote, and was reported to have said. Currently, the best selection of Cromwell's writings and speeches is still the 4 volume edition by Abbott (Harvard UP, 1937-1947). In the entirety of the published collection, Thomas Cromwell is mentioned only three times. The first time is in the introduction, laying out OC’s pedigree (which I will return to). The second is a letter from OC to his friend Robert Jenner on the Committee for Compounding in support of his cousin, Thomas Cromwell, Earl of Ardglass. The reference to Thomas Cromwell, Earl of Essex, is just to illustrate how Oliver knew Lord Ardglass (Writings of Cromwell, 1:417; Calendar for Committee of Compounding, 5 volumes, 2:950-951).
The only real reference in all of Cromwell’s known words regarding his great-uncle are those printed in a Dutch pamphlet, in Amsterdam in 1653 (Writings, 2:623-624). The pamphlet begins with a reference to OC’s “great grandfather” is the allusion to Thomas Cromwell. The pamphlet, which purports itself to be the text of an actual speech Cromwell gave in Parliament, states that Charles I was executed (in part) as revenge or justice for Henry VIII’s execution of Thomas Cromwell. Abbott states that the pamphlet is almost undoubtedly false, and I see no reason it should be taken more seriously now. Perhaps it is possible OC mentioned TC during a speech to Parliament before its dissolution eight weeks later, but it is highly unlikely that he would mention TC’s execution as justification for killing Charles. Furthermore, there is no other point in Cromwell’s career that he mentions TC and there is no mention of the speech and its contents in any of the available parliamentary or news accounts in England.
Lastly, Oliver Cromwell was not a direct descendant of Thomas Cromwell; TC was OC’s great-uncle (See below). Oliver Cromwell was a descendant of Thomas Cromwell’s sister, Katherine. Katherine Cromwell married a Welsh lawyer names Morgan Williams and became Katherine Williams (ODNB “Thomas Cromwell”). Morgan and Katherine’s son was named Richard Williams alias Cromwell. Richard was trying to create a greater air of authority and influence by keeping the Cromwell name as part of his name (Little, ed. Oliver Cromwell: New Perspectives, pp. 169-170). Richard Williams, alias Cromwell, acquired a significant amount of land from the Dissolution of the Monasteries, and it was through buying up the old monastic lands that the Cromwell’s gained their estates and wealth. The family changed their name from Williams to Cromwell after the death of TC in 1540, in what historian John Morrill called “defiant gratitude” (defiance towards Henry VIII and gratitude towards TC). Cromwell was aware of this change, and at times had referred to himself as “Williams alias Cromwell.” Cromwell may have known his family history, but he does not seem to have taken either an interest in TC’s or any record of such an interest has been lost to time.
Pedigree – Katherine & Morgan William -> Richard Williams, alias Cromwell -> Henry Williams, alias Cromwell -> Robert Cromwell -> Oliver Cromwell.
The Dutch pamphlet is called “Olivier Kromwels Oratie gedaan nade zee-slag tusschen de Nederlandsche Republijke ende die van Engeland, den 6 maart, 1653, over dese hedendaagsche en schadelikke oorlogen. Na de copye gedrickt tot London.” From the Dutch translation I could find online, it looks like it claims to be the text of a speech about the Anglo-Dutch War.