You could try these guidelines:

1) Identify the key. It is really annoying to have someone confidently bashing out a 3-chord trick in the wrong key, or who has identified the key but plays major chords when minor is required. In particular, be sensitive to tunes which change keys or which switch from major to minor.

2) Identify the rhythm, and be aware of subtleties. Just because tunes are in the same time signature doesn't mean they have the same rhythm. A tune in 3/4 isn't necessarily a waltz, for example. 3/2 isn't a slow 3/4.

I know a guitarist who appears to be unable to hear the difference between major and minor keys, and who uses the same strumming pattern pretty much regardless of the rhythm of the tune. He thinks that just because the strum comes on a beat he's playing in time. Both habits are very annoying!

3) Look for patterns. Most British and British-derived American tunes follow a simple pattern of 2-bar phrases, repeated with slight variations. Typically:

A part:
Phrase 1a
Phrase 2a
Phrase 1b (possibly with a slightly different ending)
Phrase 2b (usually with a slightly different ending)

B part:
Phrase 3a
Phrase 2a
Phrase 3b (possibly with a slightly different ending
Phrase 2b

Once you understand the pattern of the tune, you'll find you only have to learn a few short bars in order to get by.

4) You don't have to play everything. If there's a bit of a tune you haven't quite got, just miss that bit out. If you can't get your head around a tune, take a break.

5) Until you've got it, play quietly.

6) Record the session in order to learn and practice tunes

With experience, it's quite easy to fit into a new session even if you don't know the tunes, provided they're in a genre you're familiar with, and comfortable with.