Start up Nxt on your local system in Testnet mode and display peers (on the Settings Menu (gear graphic) in the header bar). The peers with a domain name are likely to be public nodes. Here are a few:

nxt.noip.me, raspnxt.hopto.org, node1.forgenxt.com

For example, put node1.forgenxt.com:6876 in your browser to open the Nxt Wallet.

Some of the non-domain name (IP address only) will work too.

You can also look at the nxt.defaultTestnetPeers setting in nxt-default.properties However, if you are planing to make changes to the client it makes sense that you run your own testnet node on your development workstation and download the testnet blockchain first.