After you close you're pretty much free to do what you want, just it'll vary a bit based on whether that new mortgage tradeline is on your report or not and that could go either way with your individual credit profile.
Personally I wouldn't put two big ticket purchases that close together unless I had no choice; certainly if they knew about it, your mortgage lender would be pretty annoyed to see the money you claimed as a reserve vaporize with a car downpayment. My non-credit, financial defensibility recommendation: don't blow your excess cash on a new car after just buying a house. There's almost always going to be an unbudgeted expense or two, and it's better not to get caught flat-footed.
Also be aware that your new mortgage won't appear on your CRs immediately, it could be 2 or 3 months, depending the creditor.