So your aim to use this TL-WR702N is that this Nano picks up the public park wifi first and it also passes its wifi signal out to allow other wireless devices to share wifi? If what i guess is right, i think what you need is a wireless router which supports WISP function.
BTW, Is the park wifi free without any charge? When you connect to the park wifi, do you need to pass through the authentication process, similarly to a hotel wifi?

This Nano TL-WR702N doesn't support WISP mode.

Matthew cook is right, you have to go on a website and pay £18 for the wifi there.

I couldn't really explain what I want to do, so I drew a picture.

Yes, bridge mode is what you want - does exactly what you want: http://www.tp-link.com.au/article/?id=394

Well, I got here, and set up bridge mode and now my blackberry and samsung and playbook all can't say that they failed to obtain an IP address.

Any idea what I have done wrong ?

okay, i got a step further. Now, the connection works, but, it only lets you see the log in page for the internet provider, nothing else??

Any ideas?

I was looking at doing something very similar to what you're after, and see something similar too.

The bridge just passes everything across, so your wireless devices are getting IP addresses via DHCP from the main WIFI network (the one your laptop is connecting to). Hence as far as that network is concerned, the devices are just connected to it and so it is dealing with them normally (ie serving up the log-in page).

What I found whilst also trying to solve this dilemma is Virtual Router - it's a bit of software which installs on the laptop and you can then use that to share your wifi link to the other devices. The key difference (at least as far as my testing shows) is that only the laptop gets an IP address from the main WIFI. The software assigns the IP address for the wireless devices which I think is what you want.

Of course the laptop does have to be connected and the software running for it to work. Also I doubt the range will be too much as it's essentially an ad-hoc network set up by the laptop wifi card. However from my test (using a Win7 laptop and a Nexus 7) it seems to work ok.

And of course you could always then use the 702N in either bridge or repeater mode (which seem essentially to be the same other than whether the SSID is selectable or cloned) to boost it if the laptop card range isn't enough. The software will create its own named network (you set the SSID in the software, along with WPA2 password) and you can use the 702N to bridge/repeat that SSID rather than the main one that the laptop is connected to.

To be honest I gave up as holiday was a nightmare as the dog got the shits and poo'ed on the caravan floor. Not sexy.
However, I did find a few bits of software that worked. But, I don't like the laptop being on for long periods of time. infact, I don't like the laptop. The mouse pointer is too difficult to see.