I had the Galaxy Nexus and I still consider it one of the best phones that I owned. I still have it here as a backup device for my GS3. Both devices are great in their own way and both have their flaws. Anyway... time to answer your questions.
The battery life on the Nexus was its weak spot, I will whole heartedly agree with that. With the 1850 or 2100 battery the battery life never came close to the GS3. I love the battery life on the GS3 compared to the GNex but its all relative. Yes it drained from 100-90 insanely quick but once it got to 89 it drained much slower and seemed to even out. With phones, I am perfectly happy if I can unplug my phone from the charger in the morning and have enough to make it through a full day and plug it in at night when I go to sleep and both phones have allowed me to do that. GS3 gave me extra juice at night but the Nexus was on life support by then lol.
I have never had an issue with signal on either phone. I will say though for sure that the GS3 handles the 4G 3G handoffs much better and quicker. I have tried the new Jelly Bean radios on the Nexus and they have improved the signal issues but I still think the GS3 is better in the signal reception area. This will all come down to where you are during the day obviously but if you are in good signal areas the majority of the time then you wont have any issues at all. If you don't get 4G in an area and you are going to in that area for a while (like you mentioned work) I would turn off 4G for that period of time and you will be fine.
I have flashed Jelly Bean on my GS3, running AOKP B3 and loving life. I love this phone with Jelly Bean on here but I agree with Cyber Warrior on that I could stay with the stock OS ROM on here and be completely happy. This stock TouchWiz ROM is great, but I just love JellyBean and while it is still buggy on the device there are many quick fixes that improve the performance. Battery life has been slightly effected but I am still getting more than a full days use out of the phone.
Either way, you have a good phone in your hand. Personally I would take the GS3 at this point but the GNex isn't that far behind especially with Jelly Bean now.