Were contractions used in spoken English in 18th-century London?

Shakespeare's texts are full of contractions, as are Dickens's:

Georgiana looked from her wine glass at Mr Lammle and at Mrs Lammle; but mightn't, couldn't, shouldn't, wouldn't, look at Mr Fledgeby.

It seems unlikely they died out in London between the lives of those two writers (both of whom spent the bulk of their writing lives in London). According to Patricia O'Conner and Stewart Kellerman, contractions were sufficiently in use during the 18th century to have purists rail against them. Jonathan Swift ranted about their use in The False Refinements in our Style. So I think you can confidently use contractions in your coming work; it's more a question of choosing the right ones for the period.

The Life and Opinions of Tristram Shandy, Gentleman by Laurence Sterne, was written in the middle of the 18th century, more than a century before Dickens's novels and a century after Shakespeare's plays, so it's more to the point:

Chapter 1, vi: Or, if I should seem now and then to trifle upon the road,--or should sometimes put on a fool's cap with a bell to it, for a moment or two as we pass along,--don't fly off,--but rather courteously give me credit for a little more wisdom than appears upon my outside;

Chapter 1, viii: and for the rest,--why--God speed them--e'en let them ride on without opposition from me;

Chapter 1, xii: that when he thought, good easy man! full surely preferment was o'ripening

Most seem to be there for 'tis and 'twas and to elide the e in past-tense endings, as in mix'd so that it isn't pronounced as a two-syllable word.

While they were there, perhaps they weren't as popular as they are today. You might want to check other 18th-century novelists and playwrights' works. Making assumptions based on the work 17th- and 19th-century writers doesn't seem to me to be sound or scientific practice. And if you're called on it, and find yourself pointing to Dickens and Shakespeare, you'll turn crimson (well, I would) if your critics point out that they weren't writing dialogues typical of the speech of Londoners in 1795.