+2
Can we say 'Nice to meet you' even before giving our name?

e.g.

A: Nice to meet you I'm Allan.

B: Nice to meet you too. I'm Jim.
+0
Yes, it sometimes works out that way in the heat and confusion of introductions.
Comments  
Teachers: We supply a list of EFL job vacancies
so under normal circumstances would that sound a bit awkward?
No, I wouldn't consider it so. It is just a matter of how the conversation and handshakes flow.