This is essentially a cluster randomized trial (assuming you randomly assigned treatments to replicates). There's plenty of literature on that (e.g. this book) and probably also in your field with terminology closer to what you are familiar with, but I do not know what you would have to search for there.
If you had a larger number of replicates, then a GLMM for binomial data (random effects logistic regression) with a random plate effect on the intercept and a fixed treatment effect would be a good option. With so few replicates, that may not be such a good idea, because it is hard to estimate the between plate variation. If there are previous experiments you could go Bayesian and use prior information on this nuisance parameter - in fact if there's lots of control plates watched over a similar time period, I would be tempted to use that as more prior information about a presumably very low death proportion in control plates. This is almost a poster child example for when Bayesian methods would be helpful - let me know when you publish this somewhere, it seems like a good examples to use.
Instead, for a traditional frequentist analysis you may need to analyse this data by summarizing the results for each plate by a single number and comparing the 3 numbers per treatment. To minimize assumptions a non parametric test might be interesting, but again you have very few data points, so the test could probably never be significant at the 0.05 level despite the seemingly very clear results. A parametric test (e.g. t-test) may require a transformation of the data. A logit transformation is relatively popular, but does not cope with zeroes unless you add a continuity correction (add 0.5 deaths in 1 additional worm). Alternatives could be arcsine square root (arcsine of the square root of the proportion that died), but no matter what you do you'll probably not get around the problem that you have no variability in the control group, so people might be worried that the assumptions of most parametric tests might be violated. I'd guess that they are not really with a suitable transformation, so a t-test of transformed (with the transformation chosen so that it's well known in your particular field to minimize reviewer comments) data is probably the least controversial traditional approach.