In my opinion it will definitely be quite a few degrees colder in late October than at the beginning of the month.
I always used to go backpacking in Snowdonia (again not the Lakes, sorry!) in the first week in October, because that was the earliest my then employers would allow you take your autumn holiday. Weather used to be somewhat cool, sun and cloud, and heavy rain usually during the night. Had frosts on one trip, four days of very strong wind on another.
Think about what it's like towards the end of October, heading up to fireworks night. You've also got shorter days to consider, too.
Later on I applied to go at the end of September, the boss didn't say anything so that became my usual date. Now that I'm retired I can go any time in September, but I've had much more wet weather in the most recent years. Climate change, I reckon.
