You're Nan sounds like an amazing lady.
Re. what to say to your DD, Winston's Wish has some very useful info on their website. We had to try and explain death and bereavement to DD1 when she was about 2 3/4 (the baby we were expected died in utero at 36 weeks). IIRC the advice was very consistent in that its best to speak plainly and truthfully and to try and avoid euphemisms - young children are very literal beings and if someone tells them "Nana's gone to sleep and this time she won't wake up" they can become terrified of going to sleep for the fear they won't wake up.
DD1 took it very pragmatically - she came and saw her baby sister in hospital and came to the funeral and burial, all of which she coped with absolutely fine. She talks about her baby sister from time to time - sometimes in a very matter of fact way (I'm now pregnant again - there's been a fair amount of "Mummy, if the baby in your tummy doesn't die this time I thought it could use the pushchair you used for me." style comments) and other times she's got quite upset. We've just tried to be there for her and help her deal with her emotions as best we can - and if that means we all wind up in floods of tears from time to time then maybe that's a good thing!
I don't mean to compare my IL's dog with your Nana, but we were also staying with at the IL's house when the dog had to be put down recently. DD had seen the dog getting older and starting to find life harder, so we took the approach that as animals get older their bodies have more problems and start to wear out - and at some point the animal may be suffering by trying to carry on living in a very old body and at some point the body may no longer be able to keep working so the animal will die (although in this instance the vet was involved and "helped the dog to die gently"). In those circumstances, death can be a blessing as the suffering comes to an end.
I think what DD's struggled with the most is what happens to the "person" when they've died but their body's still here - for us some variation on a theme of life after death (even though we're not particularly religious) has helped us find an explanation that DD seems to accept and understand. Waterbugs and Dragonflies was recommended to us by our Rector (we had a church service) and we didn't find it overly religious, plus for us it followed up on themes that the rector used in the service so dovetailed in quite nicely to give DD a consistent reference to try and fit the bereavements into.
Sorry - that's all got a bit waffley. I truly believe that both your 3yo and your unborn baby will cope with the bereavement - your grieving will not harm either of them and will help you immeasurably. Being sad when someone dies is normal and allowing your 3yo to see that is more likely to help her cope with her own grief than cause problems. I hope your Nana passes on as peacefully as is possible, you get support from your family and friends and you can treat yourself gently over the next few weeks and months.