I think it depends in part on which state you live in, different states require different procedures. In Georgia it takes about 30 days. Of course that's if you're actually refinancing the house, which is pretty much like taking out a regular home loan. If you're just doing a home equity loan I'm guessing that would be quicker.
My refi went fast, basically a couple of weeks to do, and got to skip a month on payments as the paperwork was transferred.
I had an FHA loan which is for first time buyers. I did the .gov FHA Streamline loan back when rates were at 5.5% I was paying 7.875%.
No closing costs, no out of pocket expenses, and best of all, I didn't have to get a new credit report run or even fill out any financial info.
Wells Fargo bought the loan and their site is very easy to work with on setting up online payments and schedules. And I am saving a lot of money at the lower rate.
You say your house is worth $450k. How much do you owe on the 1st mortgage? What state are you in? If you do a home equity line of credit or loan it can be very quick with no out of pocket expeses. Variable rates in the low 5's and fixed between 6 to 7%. If you refi instead of going with a home equity product there will be costs you will have to pay, but you will get a better rate.