Other than the process of liquid liquid solvent extraction, how can gadolinium be separated from scrap mixed oxide of uranium dioxide and gadolinium oxide?
If you do not want to dissolve the particles, probably your best bet is magnetic separation. Gd2O3 is paramagnetic with a relatively high magnetic susceptibility, common property among rare earth compounds (Wikipedia gives a value of +53,200·10−6 cm3/mol). An electromagnet is needed. Better, you may want to befriend somebody who has an induced magnetic roll magnetic separator; this is common equipment in ore processing, incorporating a strong electromagnet, and lab-size units are available.
For a brief description and picture of this tool, see https://www.bunting-redditch.com/magnetic-separators-for-mineral-processing/.