Sorry, as I was typing Pepper was trying to climb on the table to groom Sooty (our 22 year old cat) and managed to post too soon
Anyway, as I was saying, Pepper has been amazing with the cats from day 1 and they are really relaxed with her too - they walk up to her and stick their faces under her nose to get a groom, and she climbs into their beds to sit next to them while they sleep. So it is totally possible to have really happy puppies and cats together.
One of the cats isn't massively confident and is wary of one of our other dogs, who does chase her. We were really firm from day 1 that Pepper couldn't chase the cats, and if she got too much for them we always distracted or stopped her rather than taking the cat away.
I would be tempted to get the cat first so you aren't introducing a new type of animal into the dog's environment. If you wanted to get both at the same time, could you maybe look for a rescue cat that has previously been happy living with dogs?
Sorry to hear about your old cat, and best of luck in your search for your new animals.