Lord Crewe bore multiple titles by the time of his death in 1945. At the time of his death he held the peerages of Marquess of Crewe, Earl of Madeley, and Baron Houghton.
You have to investigate each of these peerages do determine how they were inherited. Many titles were created by letters patent, and this document would stipulate at the time of the title's creation how it is to be inherited.
Lord Crewe was created Marquess of Crewe and Earl of Madeley on 3 Jul 1911. In the London Gazette (18 Jul 1911) is the notice:
Whitehall, July 17, 1911.
The KING has been pleased, by Letters
Patent under the Great Seal of the United Kingdom of Great Britain and
Ireland, bearing date the 3rd instant, to confer the dignities of Earl
and Marquess of the said United Kingdom upon the Right Honourable
Robert Offley Ashburton, Earl of Crewe, K.G., and the heirs male of
his body lawfully begotten, by the names, styles and titles of Earl of
Madeley in the County of Stafford and Marquess of Crewe."
At the time of his father's death in 1885, Robert, being the only son of Baron Houghton, inherited that title. His father was the 1st Baron Houghton, the peerage having been created in 1863. In the London Gazette (7 Aug 1863) is the notice:
Whitehall, August 7, 1863.
The Queen has also been pleased to direct
letters patent to be passed under the Great Seal, granting the dignity
of a Baron of the United Kingdom of Great Britain and Ireland unto
Ricahrd Monckton Milnes, Esq., and to the heirs male of his body,
lawfully begotten, by the name, style, and title of Baron Houghton, of
Great Houghton, in the West Riding of the county of York.
As you can see, to inherit the titles Marquess of Crewe, Earl of Madeley, or Baron Houghton, one had to be both legitimate and male. Your grandmother, being neither of these, had no claim, therefore you have no claim to the titles. Since Lord Crewe had no surviving sons at the time of his death, the titles became extinct.
The titles aside, biological relationships did not necessarily translate into a legal relationship. For example, in England the child of a married woman was legally presumed to be the child of her husband, whether he was or was not the biological father. An illegitimate child born after the 1874 Births and Marriages Registration Act could only have his or her father named on the birth certificate if the father was present at the registration.
It may or may not be true that your grandmother was daughter of Lord Crewe – I haven't seen the evidence. I somehow doubt that he is named as the father on the birth certificate. I suspect public figures were fairly frequently accused of fathering illegitimate children, and I'm sure sometimes it was true. I have a case from about 1700 where my relative accused a well-known community figure of being father of her child – and having read pages and pages of contemporary court documents, I doubt it was true. It may be that the only way to definitively prove such a relationship is with DNA testing. I only say this because though it may be exciting to discover you are descended from an Earl, many people have claimed to have nobility in their ancestry without having solid evidence to back it up.