You will notice the difference in your shower and your laundry almost immediately once the system is in. The work usually takes a pro about one full day to install, meaning you won't have to live in a construction zone for long.
Late winter is your best bet to book this. Plumbers are often slower then, so they might be more flexible with their schedule and pricing. Get at least three quotes before you commit to anyone.
Remember that the 0% intro APR is just a temporary grace period, not a free loan. If you still have a balance left when the 15 months are up, the card will start charging interest on whatever is remaining.