First of October, Lake District then onto Ireland what to pack?

I'd be thinking about the lovely local wool hats, sweaters, shawls and scarves you can buy when you get there. Probably not inexpensive but will last and bring fond memories.

Good plan. Love the Lake District. We stayed at Keswick.

For the Lake District and Ireland, take a good rain coat and hat.

Also take rain-resistant footwear for the Lakes. It rains in Ireland, too -- that's why it's green.

Bring dry shoes (e.g. trainers) for after walking in the rain. Nice to have dry shoes for visit to the pub. On heavy rain days my "waterproof" boots don't keep my socks dry. I also carry an extra pair of dry socks for mid-day changes during wet weather.

Do you carry rain pants and gaiters? Gloves?

Great ideas all around, especially carry extra pair of dry socks. I do have rain pants!

Thank you
Stephanie

For the Lake District, waterproof your shoes as much as you can. This time of year you could have a lot of rain and, consequently, mud - dirt mixed with what the sheep leave all over!
If you're lucky, and depending on the weather, you may see the farmers gathering the sheep from the fells to bring them down to the farms for the autumn and winter - you're a little early but if it's cold enough it may happen. I don't know when exactly you'll be there but there is a local shepherd's meet happening on October 12th at Wasdale Head. It's one of the smaller agricultural shows and almost entirely local farmers. These shows are amazing to witness and a a slice of England that is becoming more and more rare.
Most lodging there will require to leave your boots outside. Some will have you take them off outside and then set them on a tray in your room to protect the carpets. I usually bring plastic bags to store my shoes in - they won't dry between the Lakes and then traveling around Ireland.
The Lakes are very casual! Hiker and farmer chic. You will only see people truly dressed up in dining rooms of the finer hotels - even in restaurants the most I've seen people dressed up would be equivalent to "smart casual".
You're going at a great time...tourists are thinning out for the season and the weather could go either way...if it's clear it will be blue skies and puffy clouds. If not...well, rain. Either way it's the best place in the world.

Our hotel had a drying room down the hall where you could hang wet jackets and try to dry boots. We also asked for newspapers (which they seemed to have on hand), and stuffed our boots at night. In the morning they were reasonably dry. In both England and Ireland we experienced downpours and clear sunshine all within an hour. Be ready for anything. It's lovely!

Yes, B&Bs often had stacks of old newspapers to use for drying boots. Pull the insole, cumple up balls of dry newspaper and stuff them into the boots to absorb moisture. Change to dry newpaper balls as needed.