Would it be possible for a common class person to teach chess or sciences to nobles during the Industrial Revolution? I think higher classes would be disgusted by the idea of a commoner teaching them, but my history knowledge is rusty. Was there a known precedent for this (i.e. lower-class instructors for the wealthy)? Would such a person have a chance to speak as equals with the noble in rare occasions?
Chess and science are two totally different.
In any case, if you are talking about England the answer is no. The only reliable way to make money at chess in England in those days was to play in coffeehouses for small stakes. The pre-eminent example of this type of player was Joseph Henry Blackburne. The general social conditions in those days would the likelihood of somebody starting off as a chessplayer and ending up as an advisor to the high and might extremely unlikely to say the least. In Germany chess was more popular and there were regular clubs and teams so it was considerably easier for someone to make a living at it there, but even so, chess was no path to the upper classes by any means.
Also, in 1880 the distinction between "nobles" and the well-off was decreasing significantly.
Social mobility during that time was low. So an individual would have to be pretty special to be noticed.
For something like teaching one would need references to be accepted by the upper class. Or acquire the endorsement of a pillar of society, preferable someone of nobility. An example may be boxing. It started out as a rough lower class amusement but after the Queensberry rules in 1867 is was adopted by the upper class.
So conceivably a good lower class boxer might build a reputation, be noticed, sponsored, and end up as a teacher of some upper class hobbyist.
I can't speak directly for England, but will note that it was much more socially liberal than Austria-Hungary, where the following exchange took place:
As recorded in Irving Chernev's "The Bright Side of Chess," a Mr. Epstein, head of the "securities" (stocks and bonds) exchange, was taking chess lessons from Wilhelm Steinitz, then champion of the world. One day, in discussion their various statuses, Steinitz announced "On the Exchange, you are Epstein and I am Steinitz, but over the chessboard, I am Epstein and you are Steinitz."
Basically, Steinitz was claiming "equality" with the much higher-ranking Epstein, based on his mastery of chess. According to comments on other posts, Steinmetz also taught rich and high ranking people in England, as well.
More to the point, most "nobles" or high ranking members of "society" are happy to mingle with people at the pinnacle of some activity that they are interested in. That is to say, a noble who was interested in chess would be happy to have the company of a "noble" of the chessboard. Most nobles would be more interested in learning something (of interest) from someone who was a clear "master" than they would be disgusted by mingling with a non-noble. In a sense, one of the advantages of being a noble is to have this kind of access to others who are "tops" in their fields.