Thanks, everyone, for your replies! After much deliberation, we are going to go ahead and do the drive. There's no direct train from Southampton to Edinburgh, and I think trying to wrangle all of our luggage, plus the fairly massive cat carrier (did I mention he's big? we think he's part ragamuffin) onto a train, PLUS a transfer would just be waaay too difficult.
I think the concerns that you guys brought up are completely valid, so we're going to look into possibly breaking the drive up into two days, and also definitely reading up on highway codes and signage before we arrive! Oh, and we will, of course, be investing in a good atlas!
I'm sure it will be a little frightening at first, but my husband apparently did some driving when he was living in Thailand (where, and this was news to me, they drive on the left side as well), I've done some driving on mainland Europe, and we've both been living/driving in northern NJ and NYC for the past five years or so, so we at least are familiar with some other stressful driving environments! I know those experiences are all going to be different than driving in the UK for the first time, but hopefully (knock on wood, head, any other hard surface available) our excessive amounts of paranoid preparations will translate us into being even more hyper-aware and safe than normal. (Bmore_2_UK: I can totally see us having a similar experience to yours, and concentrating so hard that we are exhausted after a few hours...)
I just looked quickly at Google Maps, and it suggested the following route: A34 --> M40-->M6-->A74(M)-->A702. I'll obviously be spending some more quality time with the atlas to figure out if this is the best route or not, but in the meantime, I'm wondering what an A type of road is like? If M roads are divided motorways, what is an A road?
Thanks again for all the tips and advice!