I can give you a general overview of how London periodicals and books would make their way to outlying areas, and some context on the state of medical publishing in the second half of the 19th century.

To start with the second half, medical publishing was all the rage in the 19th century. By the 1870s, the Lancet and the British Medical Journal were locked in a race for the title of most popular scientific magazine. David McKitterick notes that entry into the medical publishing market was considered a guarantee of profit by London publishers--the periodicals were popular with a broad audience, not just actual medical professionals. In fact, more serious medical periodicals and books, especially ones that dealt with sex, typically carried publishers' warnings to bookseller to check the trade affiliation of buyers to make sure their interests were professional instead of prurient. So basically, the Lancet was hot stuff to read--in London, at least.

London wasn't the only publishing center in the 19th century British Isles, but it was the most important by far, and it's the one under consideration here in any case. There are two basic ways a person (doctor or otherwise) from an outlying area could acquire a London periodical: order it directly from the publisher or buy it locally. Book and periodical (magazine, newspaper) publishers very quickly made an alliance with the new railroad system and the post service. There was a special discounted rate for shipping book packages that was extended to periodicals in 1856, and in fact the rate got cheaper in subsequent decades.

The other option would be to acquire the periodical from a local reseller. By the late 19th century, it seems like the more usual system for periodicals would be to ship a bulk package to a local reseller once a month (the Lancet was published weekly, so you'd get 4-5 at once). Sometimes the reseller would be a representative employed by a London publishing house; sometimes it would be a local businessman.

There is also an indirect way that a country/town doctor could learn about the hottest new scientific developments. From the 1850s, London publisher J&A Churchill printed a biannual digest of the latest medical studies.

As to the importance of London in the medical trade specifically: one of the phenomena McKitterick discusses is the contrast between the vigorous London medical publishing sub-industry--and the ongoing difficulty that non-London publishers had trying to establish their own. There's a litany of failed non-London-based journals from the later 19th century. (The cheap delivery rate, London prestige, connection to universities, wider publishing catalogues allowing London companies to set lower prices, and a prejudice against rural/Scottish/basically everything not London all contributed).

So overall, it seems quite likely that Whitby doctor with some professional pride not only would have been able to read the Lancet, but would have seen it as crucial to his career.