Is there a way to transfer my photos from Flickr to Google Drive without downloading all the data onto my computer - maybe using a Web Service?
Pixelpipe is great for transferring your images between various photo sharing sites.
The feature allows you to quickly select any photos, video or audio files that are currently hosted on Pixelpipe & export them in bulk to a selection of 20 of our most popular services.
http://www.flickrtoplus.com/ is a pretty simple 3-step online wizard which authorises to both flickr and google and copies over your sets.
https://www.flickandshare.com/ Allows you to download your sets, you can then re-upload them however you like.
https://www.picbackman.com/flickr-photo-video-bulk-uploader/ is an apt solution to migrate your Flickr photos to Google Drive or any other online account like Facebook, SmugMug, Dropbox OneDrive etc.
Here's a guide that walks you through doing it - https://medium.com/@llevar/how-to-migrate-your-flickr-library-to-google-photos-for-free-using-apis-and-python-2c97d3cb54a2