I'm afraid your father's story is utterly ridiculous, on a number of levels:
Level 1: Being aristocratic didn't get *anyone* out of the British Army in WWI. Just the opposite; it was expected that upper-class men should join the forces and lead their 'inferiors'; into battle from the front; that was what they were for. and they genuinely did; the death-toll of upper-class men in that war was actually proportionately far *greater* than for working-class men.
Level 2: And if it had been somehow an 'aristocratic name', and even if he had wanted to, the officer at the conscription depot would have had no power to exempt him on that account; the machine didn't work like that. All he could have done, and what he probably would have done if he had realised that the conscript in front of him was a 'gentleman'; (which of course is by no means the same thing as an 'aristocrat';) was to say ';Hey, you shouldn't be a ranker - off you go to officer school'. The snag with this piece of 'privilege' is that as a junior officer he would have been right in the very highest-risk rank of the entire army; a subaltern's life-span on the Western Front was statistically far shorter than that of an Other Rank.
Level 3: There have never been *any* aristocratic Whewells; no family of that name has ever held a peerage. In fact the first crest you found belongs to the only person of that name who has ever been notable enough in any way to make it into the enormous Oxford Dictionary of National Biography: Dr William Whewell, who was a Victorian clergyman and scholar. This Whewell was of very humble origins - his father was a carpenter in Lancaster - but his academic achievements brought him into the scholarly middle class and therefore entitled him to apply for and obtain a grant of arms. However, he died childless, and therefore this coat of arms died with him; nobody in the whole wide world since then has been entitled to use it. Even if you could prove you were descended from one of his brothers or sisters, for example, you still would have no claim to it: it was granted to him and to 'the heirs of his body'; alone. The full grant of arms is listed at the link, which is the official registry of British coats of arms as at 1884. This registry doesn't list the other crest you mention at all; it may well be an even later grant. But in any case, unless you could prove you were the senior lineal male descendant of this 'Thomas, Esq. of Fair Elms, Blackburn' (and you do realise, don't you, what a very modest middle-class status and address that is?) you could not in any sense claim that one as having any connection to you either, let alone call it your own.
My guess is that your family story is just hopelessly garbled. It's perfectly possible that the recruiting officer was a friend of your great-grandfather's family, and when he heard his name he either got him out of service on some pretext or other (e.g. advised him what to do to fail the medical examination) or swung it for him to go into a behind-the-lines job; of course that sort of thing happened. And that might well have gone down in family lore as 'because the officer recognised the name'. But it certainly wasn't because the name was aristocratic.
However, by all means do research your ancestry and see if your family tree joins up with that of either of those people. It might quite possibly join up with both: Blackburn and Lancaster are only about 30 miles apart, after all, so Dr William and Thomas Esq might quite possibly have been cousins in some degree. (That still won't entitle you to claim either of their coats of arms, though!)