gentlemen first and foremost for a cocktail or light fare, it really depends on the lady. There is no general consensus I'm afraid. My personal schtick is that it's kind of included in the whole encounter. The date starts when we first lay eyes on each other wherever that may be. But then again, I'm not a clock watcher to the minute, so again, each lady is different. If we're having a good time, we're having a good time, let's just enjoy it. I'm not one to have another gentleman lined up after, so that's a worry I don't instill on myself. I like enjoying myself too here in this world.  

But I have to say, having a 'quick' drink for 10 minutes is no way to process the notion of starting a date together in my honest opinion. Rushing anything isn't. If you want a non-rushed setting including a cocktail so you can enjoy getting to know one another in a social setting (which in it self is terrific foreplay) simply book some more time with her and don't sweat it. Just enjoy her and your date together. But again, each lady and her approach to this notion is different. And as far as starting your date in a social setting first? I highly recommend it.

because it requires more dressing, hair, makeup.  If you want such a date, expect to pay for the time...  she's already "giving" the extra prep time.

There is no way to know without asking her directly.  Some will consider it part of the time you're paying for, others will not.  My suspicion is that most would count it as part of your time, not off the clock.

Swim

If her preference was to meet you at the bar and you're concerned about the clock, be polite if you want to move things along or clarify whether it IS on the clock.  I hope that others (guys and gals) can make some suggestions.  

Do NOT say, "Hurry up and guzzle that down!  I'm not paying to watch you nurse a cocktail!!"
Do NOT say, "What are you doing? Waiting for the ice to melt??? Bartender! Turn off the fucking AC!"
Do NOT say, "Here.  I brought a plastic bag and a straw.  Pour that in and let's go upstairs, NOW!"

Maybe: "It seems like everyone here has their eyes on you.  I can't wait to get you away from them and to be alone with you."

Now, if she says, "Don't worry.  This is OTC." you can relax.  Other replies?  Got to play it by ear

If she has a schedule to adhere to, then your 1-hour date will only be 1 hour long, regardless of activities. For instance, when I worked for an agency, I'd very often rush from date to date...if I was supposed to meet you at 8 for a 1-hour date, then myself, my agency, and my next client would expect me to be out the door by 9:05 at the absolute latest. Even independents sometimes have two (or more) dates in an evening... or, they might have a babysitter at home, or some other reason that they cannot easily accommodate 'soft' scheduling.

If she has no other obligations, she might be able to accommodate a drink. But in my experience, a 10-minute drink rarely lasts 10 minutes. More like at least 30.  

If the drink was at her request, then in my view, the time is on her. If the drink is your idea, and social time is important to you, then I'd say to book a longer date. That way, you're covered....

You could always phrase the request in such a way that she can offer to have the drinks be OTC. But don't count on her doing that. It's much more likely to work if she knows you already, but far from definite.

As was said earlier, though, she's the only one who can give you the answer you're looking for

Thanks everyone for the replies and input.  Much appreciated.

A nicer way to "ask" would be to say something like, "Since this is our first meeting, I'd really like to have a get-acquainted drink at the bar first.  What would your rate be to add on a bit of time for that?"

Nine times out of ten, for a polite, respectful request like that, an independent escort will just give it to you.  We like things to be fun and relaxed too.

If she says an amount that's more than you'd want or be able to do, tell her.  Something like "I'm sorry that's more than I can do- let's just make that time part of the hour."  And then let her run the hour.  If a hobbyist has a different/bad experience, please chime in, but with me, I'd say do not try to rush to maximize the use of your hour.  If you spend half an hour in the bar, a good provider will work doubly hard during the second half hour or offer to let the time run over.

When women feel like you're looking after them (offering to pay for the time, not rushing to max the use of it, etc) we're more or less hormonally flawed to fall all over ourselves trying to make you happy.  Pick well, then exploit that.