There is no reason to doubt that Churchill explicitly mentioned Rolls-Royce in the context of the quote.
A Google Books search limited to books published before 1960 returns at least four publications that feature the Churchill quote including the Rolls-Royce reference:
- Jackson, G. G. (1930). The world's aeroplanes and airships, p. 173.
- IPC Transport Press Limited (1944). Flight International, p. 639.
- BBC (1954) London Calling, p. 12.
- Wallace, G. (1955). The Flight of Alcock & Brown, p. 302.
[…] their science, their Vickers-Vimy aeroplane, their Rolls-Royce engines – or their good fortune. All these were necessary – and all of them contributed […] The world's aeroplanes and airships
When one considers all the factors I really do not know what we should admire the most in our guests – their audacity, their determination, their skill, their science, their Vimy-Vickers aeroplane, their Rolls-Royce engines, or their good fortune. All these were necessary, and all of them contributed to their achievements, and to the event which has brought us all together here this afternoon, to cheer the victors of the first non-stop Atlantic flight. Flight International
"I do not know what we should admire the most," he said, "their audacity, their determination, their science, their skill, their Vickers Vimy aeroplane, their Rolls Royce engines, or their good fortune; for whatever we should admire most we are celebrating their stupendous achievement which has done as much as anything in history to add to man's growing understanding of the air." London Calling
Google Books doesn't show me the 1955 quote, and there are some differences in wording. However, all three quotes that I could check agree in that according to them, Churchill explicitly mentioned "Rolls-Royce engines".
[EDIT] As the comments reveal, the OP is a skeptical mind indeed who is not easily convinced that the whole quote is not a fabrication even though it is cited for instance in a BBC publication. Thus, he asked for a 1919 source that confirms that Churchill really praised the contribution of Rolls Royce to the first transatlantic flight. Personally, I was already happy with the Google Books results, but I did go the extra mile and checked the Times Digital Archive for their coverage of the event.
And indeed, I found a transcript of the speech that Churchill gave on occasion of the ceremony that awarded the Daily Mail price for the first direct flight across the Atlantic. The speech was published on Saturday, June 21, 1919 on page 7 (link may be pay-walled). As it is, the transcript is word for word identical to the quote printed in Flight International reproduced above. So, all this confirms that yes, Churchill did indeed praise a British engine manufacturer for their contribution in the first transatlantic flight (unless we are so much of an skeptic that not even this quote from 1919 can convince us that it's not made up). [END EDIT]
Incidentally, the name of the aircraft "Vickers Vimy" is also explicitly mentioned in these quotes, but it is missing in the plaque that is shown in the Rolls-Royce promotional video that you've linked.