First, … Secondly, … &c

Style guide guidance

The strongest proponent of firstly I've encountered is H.W. Fowler, Modern English Usage (1926):

First(ly), secondly, lastly. The preference for first over firstly in formal enumerations is one of the harmless pedantries in which those who like oddities because they are odd are free to indulge, provided that they abstain from censuring those who do not share the liking. It is true that the Prayer Book, in enumerating the causes for which marriage as ordained, introduces them with First, Secondly, Thirdly; it is true that firstly is not in Johnson; it is true that De Quincey labels it 'your ridiculous and most pedantic neologism of firstly'; but the boot is on the other leg now. It is the pedant that begins his list with first; no one does so by the light of nature; it is an artificialism. Idioms grow old like other things, and the idiom book of a century hence will probably not even mention first, secondly.

But in the United States, at least, the boot moved in an unexpected (to Fowler) way—supporting not firstly in place of first, but second, third, and last in place of secondly, thirdly, and lastly. Here is Bryan Garner, Garner's Modern American Usage, second edition (2003) on the same subject:

firstly, secondly, thirdly, etc. are today considered inferior to first, second, third, etc. Many stylists prefer first over firstly, even when the remaining signposts are secondly and thirdly.

From Kenneth Wilson, The Columbia Guide to Standard American English (1993):

firstly, first (advs.) Both firstly and first are adverbs, and some have argued that firstly (which has been around since the sixteenth century) is unnecessary and ought to be avoided. But both are Standard, even though firstly may seem a bit old-fashioned. Remember though that in a series of items, you should stick with the form you begin with: if first, then second, and so on; if firstly, then secondly, etc.

From Allan Siegal & William Connolly, The New York Times Manual of Style and Usage (1999):

first, firstly. Use first, not firstly, in enumerations.

...

second, secondly. In enumerating causes, factor, etc., use second, not secondly. The phrase is a short form of what is second.

...

third. thirdly. Use third, not thirdly, in enumerations. The phrase is a short form of what is third.

From Theodore Bernstein, The Careful Writer (1973):

FIRST, FIRSTLY

Over the years there has been much firing back and forth over these words, but one gets the impression that the antagonists have been firing blanks. Some have objected for obscure reasons to the word firstly. Yet all have sanctioned secondly, thirdly, fourthly, plus what is almost invariably designated as etc. Whether the etc. covers forty-thirdly or eight-hundred-seventeenthly nobody ever says.) If there is going to be quarreling over whether to use first ... secondly or firstly ... secondly—although heaven only knows why there should be—perhaps the obvious and simplest way to handle a series is first ... second (both of which words, by the way, are as much adverbs as are firstly, secondly). This solution, incidentally, takes care of forty-thirdly. Surely a first ... second listing is in no way inferior to a one ... two listing, about which no one has ever fired a shot.

And from Webster's Dictionary of English Usage (1989):

Almost universally, they ["the commentators"] admit that firstly is all right, but they still prefer that you not use it. They want you to use first instead because it is shorter. Whether or not the commentators' advice has had any effect, first is a much more common and much more generally useful word than firstly, which is almost never used except to begin an enumeration. Our evidence also suggests that firstly is more frequent in British English than in American English.

... A good many recent handbooks feel that you should be consistent in your enumerations. They prefer first, second, third, etc., but will allow you firstly, secondly, thirdly, etc. if you really favor those. Now there is nothing wrong with consistency—it is desirable—but it is only fair to note that writers and speakers have often played fast and loose with these enumerators. [Examples from Jimmy Carter (1970s), Thomas Gray (1758), William Hazlitt (1808), Fitzedward Hall (1873), and Colin MacInnes (1957) omitted.] So while we do not suggest you be purposely inconsistent, it does appear that consistency in this specific usage has not always had a particularly high priority with good authors.

WDEU's handling of this topic is needlessly sarcastic and argumentative. The book seems to endorse a consistent handling of ordinal enumerations, but it goes out of its way to portray "the commentators" as behaving with ludicrous grandiosity in favoring the same thing. As for WDEU's examples of first, secondly, thirdly, etc., and other inconsistently framed enumerations, those involving the succession first, secondly, ... were precisely the ones that Fowler pointed out ninety years ago as retaining the support only of pedants. So when WDEU says "it is only fair to note that writers and speakers have often played fast and loose with these enumerators," you might ask "fair to whom"? And the answer would seem to be "defenders of inconsistency on pedantic grounds."

Google Books search results

More interesting to me is WDEU's comment that "Our evidence also suggests that firstly is more frequent in British English than in American English." Although WDEU doesn't pursue this observation with any further analysis, it seems likely to me that British English is more apt than U.S. English to use firstly in enumerations because it is also more apt to use secondly, thirdly, and lastly. To test this possibility, I ran Google Books searches for firstly (blue line), secondly (red line), thirdly (green line), and lastly (yellow line) in both British English and American English across the years 1800–2008. Here is the resulting chart for British English:

And here is the corresponding chart for American English:

The general trajectories of the lines on the two charts are fairly similar, but the frequency of usage is quite different. Even in 2000, near what must be the end of a long downward slide, the frequency of secondly in the British English corpus of Google Books was .00060335, while the frequency of the same word in the American English corpus of Google Books was .00015952—slightly more than one-fourth its frequency in the British English corpus of Google Books.

That difference, it seems to me, provides ample grounds to explain why firstly is more common in UK English than in U.S. English: people are trying to present their enumerations consistently, and many more people are working with the terms secondly, thirdly, etc., in Britain than in the United States.

Conclusions

In my many years of working in U.S. publishing, I can't recall encountering any house style guide that specified firstly, secondly, etc., over first, second, etc. Many U.S. house style guides do the opposite. Further I have never worked with a house style that urged an inconsistent approach such as first, secondly, etc.

If, at the OP's invitation, we take secondly and thirdly for granted as the next ordinal forms in the enumerated series, then the only remotely plausible reason to insist on first over firstly as the first entry in the series is to perpetuate a form that (if we may believe Fowler) was already out of step with actual usage in the UK in 1926.

But why should we take secondly and thirdly for granted? In the United States, where most style guides are either silent on the question or prefer the ordinal enumeration first, second, third, etc., justifying firstly in a series that doesn't include secondly and thirdly is rather difficult to do. In that situation, firstly lacks even a historical preference based on an antiquated argument to prop it up.

Ultimately the choice between first and firstly is a matter of style, and if you aren't beholden to a publisher's house style, you may go as exotic as you like in your treatment of ordinal enumeration—adding -ly to all odd-numbered terms and omitting it from all even-numbered ones, for example. But if you aren't trying to call attention to your unique style choices, you have three realistic options: (1) use first, secondly, thirdly, etc. (because that's what your great-great-grandfather the opium eater used to do); (2) use firstly, secondly, thirdly, etc. (because in the UK lots of people gravitate toward the -ly forms, and this is a consistent way to accommodate those forms); or (3) use first, second, third, etc. (because in the United States that's the most common method—and it's a consistent approach, too).

I have used both in my papers, sometimes interchangeably. If pressed to differentiate, IMHO first feels more direct and demanding of attention. Firstly feels slightly formal by comparison.

Fowler suggests that whether you choose ordinal numbers or adverbs to enumerate your list, you should do so consistently throughout your list. I.E. you should use firstly, secondly, thirdly, etc; or you should use first, second, third, etc. Many of the adverbial forms sound odd to me, so my personal druthers are to use the ordinal numbers.