Wind correction on downwind during circling
Yes, they need to be corrected, mainly for two reasons:

1) Legally, the horizontal criteria for circling is the distance from the threshold of each usable runway (search for TERPS circling criteria if you are interested in US rules, or PANS-OPS circling criteria for the rest of the world). In other words: you need to stay within a given distance of the runway. If you have a very strong tailwind, without correcting for it, you might leave the circling domain.

2) In order to facilitate the base/final. If you have a very strong headwind (which is indeed not very likely, as it means you will land in a very strong tailwind), you will drift towards the runway during your entire base/final-turn. This means you will roll out on the final course a lot closer to the runway than you would in zero wind. This could be a problem, since your stable approach criteria usually requires you to be "wings level" at some altitude (typically 300 ft for circling).

Now, for the how, this will depend on aircraft and operator. One example is however to add/subtract half the wind component in knots from your timing in seconds. Example: Zero-wind outbound: 30 s, Tailwind component: 20 kt. Wind-corrected outbound: 30 - 0.5*20 = 20 s.

(In addition, outbound timing is usually also determined by circling height, since the higher you are, the longer you will need for your descent. A typical SOP here is to take the circling height in hundreds of feet, and multiply it by 3, to get your zero wind outbound timing. Example: Circling height: 1000 ft. 1000/100 * 3 = 30 s.)