I agree about not wanting plastic ones. I would definitely avoid cheap plastic cups because they are usually uncomfortable to hold, with hard ridges at the seams, and can split and leak with repeated use of hot water. You can get softer plastic beakers with handles in plain colours, designed for nurseries, schools, hospitals, care homes etc. It all depends how much you want the replacement to look like a proper china cup.
If you search for melamine cup or melamine beaker you find some reasonable patterned ones on Amazon and other UK websites. Make sure you don't buy tea sets designed as toys because these are not intended for use with boiling water. The adult teas sets that look like real china are only available from America, as far as I can tell. So often I think, wouldn't it be great if you could get... and find it is widely available in the USA but not in Europe.
Patterned melamine tableware as opposed to plainer picnic ware hasn't been made since the 1970's in the UK, apart from mugs/beakers. You could try eBay where there are vintage sets. Search for Gaydon Melaware or Melmex. They come in a range of colours, with white insides and coloured outsides. The styling is elegant and is now very much back in fashion.
P.S. Don't wash melamine in a dishwasher.