These operations are going on rapidly refers to the speedy rebuilding of the Temple. We may say “They are rebuilding the Temple very quickly.”

And the work is prospering in their hands: In their hands is an idiom meaning “as they work.” This clause may be rendered “and their work is moving ahead steadily” (similarly Good News Bible).

And being completed with all splendor and care: Good News Bible provides a helpful model here, saying “yet they are taking great care and doing their work beautifully.” Good News Bible begins with the conjunction “yet” to make a contrast with the first half of the verse. In the first half the writers say that the work is being done quickly, but they also want to say that in spite of the speed with which the work is being done, it is being done carefully, and the results are good. Contemporary English Version‘s model misses this contrast. This clause may be translated “but they are working carefully and making a beautiful building.”

Quoted with permission from Bullard, Roger A. and Hatton, Howard A. A Handbook on 1-2 Esdras. (UBS Helps for Translators). Miami: UBS, 2019. For this and other handbooks for translators see here.