I don't think it's impossible, but the challenges are very high. One of the big ones will be enriching the fissile materials - IOTL, that required the development of teflon in order to stand up to the uranium hexafluoride that was being processed. A plutonium bomb won't be any easier, because a) you have to get the plutonium out of the fuel rods somehow and b) once you do it's still very toxic - so much so that during the Manhattan project, immediate high amputation was considered the only practical first aid!
I think late Victorian-era chemistry was up to the challenge, so the Edwardians should be too... but they would have to be very, very careful. Because there is only one way to get it right, and lots of insanely dangerous ways to get it wrong. Ways that can contaminate the area for miles around and kill off a whole slew of your best physicists and chemists.
Once you have your bomb, of course, you have to deliver it somehow. Remember that aircraft were in their infancy back then, and the first atom bombs weighed at least 5 tonnes. It might be possible to create a short-range unguided rocket to deploy it, with a pressure fuse to detonate it at the right altitude, but it would be a huge beast because rocket fuels were also not very good. And that's another massive research project with lots of entertaining possibilities for failure. I think you're more likely to see some sort of naval weapon, perhaps a huge torpedo or a mine deployed from a submarine full of very brave volunteers.