This morning our Peugeot 306 wouldn't start.  It's a 1998, 1.4 petrol,
110k miles.

Turned the key and engine turned over fine, started very briefly before
stopping.  Tried again, kept turning over etc, but no sign of any ignition.

Battery was replaced on Thursday lunchtime and car has been driven a
number of times since then.  Last time being around 2pm yesterday
afternoon (all of 5m up the driveway to make space).

Tried a 2nd key (in case of imobiliser fob issue) - same problem so went
back to original key.

I've had this once or twice before in the 5 years we've had it, and
always thought to be imobiliser.  Removed key from ignition and then
re-insert and try again has always fixed it.

After a few more goes, I wondered if I'd entered the radio code (okay -
stupid idea!) - it asked for code - entered it.  Tried to start again
and this time, after a bit it did start, slowly and picking up a bit at
a time.  After a couple of seconds car was running as normal and drove
us to where we were doing.

A couple of hours later, it started perfectly fine again.

Any suggestions what might have been the issue?  Previously, when the
imobiliser may have not disengaged, it would always start fine the next
time - didn't work this time, and also, it seemed strange how it picked
up once it had started.

This car always starts first time, never a problem until dead battery on
2 cold mornings this week - started fine after being charged for 15
minutes, and also later in the day.  Oh, and 2-3 times I've had to
re-insert the key to start it (in 5 years, that's not too bad I think!)

Any suggestions?

Thanks

David