Making this salad ahead of time, refrigerating it and serving it cold like a traditional potato salad is ideal. But you can serve it at room temperature if you didn't plan ahead.
Wash the trimmed radishes and halve. In a large shallow pan over high heat combine the radishes, olive oil, water and a heavy pinch of sea salt. Bring to a boil uncovered. Then reduce the heat to low, cover and simmer for 5 minutes.
The radishes should be soft and tender, and most of the water should be absorbed. Remove the radishes from the heat and set aside on a towel to cool and pat dry.
In a small bowl mix together the mayonnaise, celery, shallots, dill, mustard, salt, pepper and lemon juice. Toss the radishes with the dressing and combine well.
Refrigerate until you're ready to serve.
Any mayonnaise will work, non-dairy, regular or the like. My favorite vegan mayonnaise is Just Mayo.