I'm pretty confused by your post to be honest, but I'll try and help...
So I'm confused because I don't understand how you could be on a waiting list for 3-4 months (I would have thought only Frith Street maybe has a list that long) and then they would still draw it up on the day.
Where are you getting tattood? And is it the same place each time? It's pretty poor form of them to draw it up on the day, are you sure they haven't pre drawn it or are just adding some hand draw elements? Also, if you are just getting small tattoos or designs they do on a regular basis, swallows/anchors/stars/tribal/lettering/flowers/the list goes on, then there is no real need for them to custom draw something up.
Also, this idea of having a tattoo designer is weird, your artist should be desigining your tattoo (if they are half good) as they are the person with the experience of how ink will look on different parts of the body and they will have the experience of specific aesthetic.
What you need to do is look through the following sites, look at the artist profiles, and find an artist who tattoo's in a style you like. Then call the studio and ask when they have a free spot for you to come in and have a quick chat about what you want (make sure you take reference material), see if they feel it's something that would work and get their opinion on a design. A design should never come 100% from the person getting tatoo'd, that's where crappy tattoos come from. Your artist should be inputting into the design from start to finish, they do know better than you.
All the studios I've mentioned below are top notch (I've had about 15 hours of tattooing done at Skunx and since I moved to London it's the only place I go) and I'd be very surprised if they drew up a design on the day (unless as I've said it was something kinda standard ish)
www.skunxtattoo.comwww.frithstreettattoo.co.ukwww.evilfromtheneedle.comwww.ilovegoodtimes.co.ukwww.shangrilaparlour.comwww.into-you.co.uk