Hi Mrs James Smith 66 - congratulations on your news! I was in a very similar position to you when I found out I was pregnant at 5 weeks, and was planning to run a half at 11 weeks. I fully intended to run it, but I found I was so exhausted from about 8 weeks that I just couldn't face training for it let alone running the race. Everyone is different though, so you might find you're totally fine.
I would say there are always loads of walkers (and very slow runners) at any race - if the Bath half course is still the same as when I did it a few years ago, it is made up of two laps of more or less the same route so you can always drop out if you don't feel too good.
Even if you do feel a bit ropey for a few weeks, don't lose heart. Once the fog of the first trimester lifted, I had much more energy and I'm still running (albeit short distances, very very slowly) at 22 weeks.
Thanks Run2Mum for the fittamamma tip - been struggling to find a pair of running tights with support in them and was considering the bump support option. I'm 17 weeks now and belly is getting bigger - but I'm still running, albeit 1.5-mins per mile slower than my (already plodding) normal pace. I'm happy just to get some fresh air and my muscles moving.