While there are several methods, this is the one that I use: Log in to iCloud.com from a browser, preferably a private computer. Click on the Photos icon. Go to All Photos. Select the photos you want to download from the iCloud and then click on the Download button. All the pictures you selected will be downloaded to your computer. Now, sync those photos to your new iPhone using iTunes. Note that these photos will not be synced to your Camera Roll, but to a separate folder.
On a side note, there are a few third party apps available that promise to swiftly transfer photos from your iCloud to your iPhone. Some of these need you to log in to your iCloud account from within their interface. I don't trust them, hence this extensive workaround.