TL;DR -- that grandnephew could inherit the personal (non-real estate) property of the Earl if given to him in the Earl's will, but could not inherit the peerage title and (before the mid-late 20th Century) could not inherit real estate even if granted in the will and would have taken nothing if the Earl died intestate.
Details:
Peerages are heritable by the terms of the writ or letters patent under which the Crown granted them. Illegitimacy and adoption always break the chain of inheritance. Also, if this peerage was one of the rare ones that could pass through the female line, the niece would have succeeded herself to the peerage as Countess given that the brother in question was the elder brother (i.e., the niece would have been senior to the person who in your question has just died).
Inherit property? Here it's important to divide up your time frame. Entailment of property (see below) and the legal disabilities of illegitimacy were largely done away with over the course of the 20th century.
But for the 19th century, and early 20th century, in the case of real estate, pretty much not. The bar sinister (legal disabilities of illegitimate children) and the likelihood that much of the Earl's most valuable real estate was "entailed" (subject by its original deed to limitation upon who can inherit it) combine to exclude illegitimate children from gift of real estate by will or receipt of it in the case of intestancy.
For the 19th century in the case of property other than real estate, the answer is "yes, he could inherit" by gift of the will, and no he could not inherit in the case of intestancy.
It's not impossible that letters patent could grant a specified illegitimate child a right to inherent, but I doubt it ever happened, and it would have been inconceivable to specify a general ability of a title to pass to the non-marital or non-bodily line.
Someone with the pull to achieve that would just have the person in question (or himself) created a peer in his own right. Which most certainly did happen, and famously so in the case of illegitimate children of Charles II and George II. But even those titles once created had a firm bar sinister for subsequent inheritance.