So when I read this I thought to myself: "Yeah, I saw a fairly minimal production of that show a few months ago".
And then it dawned on me that it was Scarlet Pimpernel, not Jekyll and Hyde.
Here's the link to the pics, anyways: http://www.smc.edu/theatre/images/Scarl ... et1_14.htm
If your theatre has a nice scrim, you are in luck because you really don't need any backdrops. You can just light the scrim in different ways to suggest different scenes.
If you don't have a scrim, it's kind of a big investment (but I'm sure it would be used again and again). You could probably get a way with just one backdrop: a sort of London skyline.
The rest shouldn't be to hard. The hospital can be suggested by a few hospital beds (that you can roll on and off stage). Jekyll's lab can be a cool table with a bunch of creepy mad scientist things on it. Just have the essential scenic pieces needed.