Hi Tink!
One way to find out the location of a hotel in relation to WDW is to visit their website and see the map they post there--they will certainly have one! For the Hyatt and Doubletree, go to their national sites and click thru until you get to their Orlando hotel. For westgate:
www.westgateresorts.com/wglakes.html Keep in mind, however, that 'miles from' WDW won't really tell you how long it will take you to get there; if you have to travel the main roads and you are going during the busier seasons, even a hotel 1 mile from WDW can be a lengthy car ride with all the traffic!
We'll also better be able to help you if you can let us know the dates you are planning to travel--hotel rates vary wildly depending upon when you'll be there!
I do not know anything about the Hyatt or the Doubletree, so I can't comment on those.
This past January, we stayed at the Westgate Lakes Resort. I got our accomodations for a ridiculously low price--about $49/night for a 2 bedroom suite with full kitchen etc etc because I A. booked in the 'off' season and B. agreed to do a timeshare presentation. My kids and I loved the resort so much, I actually did buy the timeshare--even though prior to our stay I firmly believed I would NEVER buy a timeshare. Not that any of this really matters--even if we hadn't gotten the TS, I would *still* have loved Westgate Lakes as much as we did. The rooms are gorgeous, the service was always prompt, we were left totally alone (meaning, no hounding by sales reps) during our entire stay, and the sales presentation was very low-key (a BIG factor in my buying in--I hate high pressure sales). You will get some contrary opinions from other folks, but I wanted to stick my 2 cents in. If you do not do the timeshare presentation, the cost is significantly higher--I believe $150/night.
It took us about 10-15 minutes tops to drive to the WDW gates from Westgate. It was about 5-10 minutes from Universal studios. The Resort has much to offer for those days you need some downtime....I think there are 6-7 pools, and activities for the kids around each pool area everyday (they give you an activities schedule when you check in)....also at most of the pools, there is a bar with happy hour specials--$1 margaritas is what they had the night my mom and I scooted over there. One of the best features (in my family's opinion) was the Live Animal Show--we saw all manner of creatures--snakes (BIG BIG BIG SNAKES!), exotic spiders, a tiger, a lion, a panther, a leopard.....you can also have your photo taken with one of the big cats for a nominal fee. It's small and intimate and you get a chance to see these critters up close, and can even touch some of them. It excited my kids to no end!
Feel free to email me if you've got any other questions about Westgate!
Susanne