I've done many walks with kids in the backpack. Also, they quite quickly started to prefer walking themselves - we climbed Cnicht when my daughter was 3, and went on to do several bigger hills including Tryfan and Cairn Gorm.
Depending on where you are in the UK, I'd suggest the following walks that I've done are back-packable in good summer conditions.
Pen y Fan from Storey Arms
Picws Du / Fan Brycheiniog
Black Mountains - most routes
The Sugar Loaf
Moel Sych / Cadair Berwyn via Llyn Lluncaws
Moel Famau
Snowdon via Snowdon Ranger or Llanberis paths
Kinder Scout from the Hayfield side
Ingleborough from Clapham
Helvellyn from Thirlmere
Skiddaw, as you mention
Grisedale Pike
Cat Bells
Dale Head from Honister
Harter Fell (Mardale) via Gatesgarth pass
Tinto
Broad Law from the Megget Stone (
https://www.walkhighlands.co.uk/maps/map1_1bd.shtml)
Ben Lomond - but note that there is one very short steep rocky bit low down on the route
Ben Chonzie
Ben Lawers
Meall Greigh by itself
Meall Buidhe (Glen Lyon)
Meall nan Tarmachan (without the ridge)
Ben Gulabin
Glas Tulaichean
Morrone
I've not done Beinn Ime but I've done the lower part of the route (en route to the Cobbler) and it could be suitable, the upper part carrying on to Beinn Ime summit also looked very straightforward.
The Cairngorms have lots of routes that could be done with a backpack, but I would suggest not bothering with the summits and just doing the approach. For example, the approach to Braeriach from the Sugar Bowl is good as far as the Chalamain Gap. The walk to Coire an t-Sneachda from Coire Cas is good as far as the corrie.
There are also several routes around Drumochter that would be possible.
My main other thought is warm clothes. In my experience children need to be very snugly wrapped up in a backpack since the yare not moving and can get cold very quickly. In warm weather, mine needed lots of water!
tim