Nobody would ever even consider that. Critically from the British POV, Hijaz was ruled by a independent British ally, however minor, whose cooperation had been instrumental in defeating the Ottomans.
There was not even the slightest shadow of a claim over the land in question by any non-Muslim country. The most valuable known resource in the area was, well, revenue from the pilgrimage and the associated prestige among Muslims (it is still the case now, for the most part), something that any non-Muslim country could hardly consider aspiring to.
And of course, everyone would be aware that it would be seen, to put it mildly, as a major irritation to all Muslims. I am not sure that general Jihad would follow in the given timeframe, critically because, among other things, there might be a lack of recognized authority with the ability to proclaim such Jihad. But, at any rate, it would be a surefire way to secure lots of headaches to colonial empires who already ruled tens of millions of Muslims, many of whom already fairly unsympathetic to say the least, for absolutely no gain whatsoever.