Long post, so there's a summary in the TL;DR section at the end.
If you are wanting to dive into some details; sometimes, I find that looking for words with the same (often rarely used) kanji can help to get a handle on nuance differences.
愈愈【いよいよ】 : The kanji here are perhaps the least helpful of the bunch as they are almost exclusively used, if ever, for this word and its antiquated forms.
However, note that there is another usage that means
more and more; all the more; increasingly.
Examples of this usage can be seen in sentences such as
The game is coming to a close, and it's the bottom of the 9th.
That nuance gets mixed in with the "finally" meaning in the sense that your expectations have built up and finally you get some closure:
Before the long-awaited final game, our coach said, "This is it!"
漸と【やっと】 : Words with related kanji are
漸次【ぜんじ】ー gradually; slowly; little by little; incrementally
漸進【ぜんしん】ー gradual progress; steady advance
in which the word "gradual" stands out. This is probably close to what an English intuition of "finally" is--a gradual build up to a "final" conclusion.
Maria Ozawa and I are finally getting married!
Note how, as compared to いよいよ, the buildup isn't necessarily of expectations. In the above example it could be hinting that everyone else was waiting for us to get married.
漸く【ようやく】 : The character here is the same as for やっと, and in fact the "gradual" nuance pretty much carries over. However, you tend to see ようやく more often in a stiffer/more business context.
The two companies finally reached a conditional agreement.
Which sounds like they have been deliberating for quite a while.
遂に【ついに】 : Probably the most common word to see this character in is with
遂げる【とげる】ー to accomplish; to achieve; to carry out
From this we would expect ついに to be used commonly in instances where, perhaps after much effort, you finally reach some sought after conclusion.
Their years of effort finally paid off and now they're rich.
Though sometimes it's not always that cut and dry, and instead of "accomplish" we get something more like "working steadily toward an end".
The long-running TV series finally came to and end.
到頭【とうとう】 : The first character here is seen often in
到達【とうたつ】ー reaching; attaining; arrival
And, of course, you recognize the second character as
頭【あたま】ー head; top
These two indicate more of a focus on the destination instead of the process/buildup of getting there. The situation came to a head and you _finally reached the end.
He finally settled down and started a family.
いよいよ ー Finally, after a build up of expectation.
やっと ー Finally, after some gradual build up (not necessarily expectation).
ようやく ー Finally, like やっと, but more business-y or formal
ついに ー Finally, after working to accomplish something or toward some end.
とうとう ー Finally, having arrived at some end.
Disclaimer: Naturally, here I'm picking example sentences here that are pretty clear, and in actual usage you don't see such stark meaning boundaries. However, hopefully, the stuff above gives a start into building your own intuition on how these words feel.
References: Several of these examples were pulled off www.alc.co.jp, though I tweaked the translations a bit. The definitions were pulled almost verbatim from EDICT on WWWJDIC.