One thing I've stopped bringing with me is a tablet/iPad, since my up to date phone has everything I need. But if OP needs iPad for work, well that's different obv!
I have the regulation size Ryanair backpack. Honestly it is absolutely unreal what you can fit in there if packed properly and all "just in case" items are left behind. I click and collect at Boots if needed for toiletries. Vacuum compression bags are great provided you use uncrushable clothes or have an iron waiting for you other end! There are vacuum compression bags now that have a little usb powered pump. I must get some....
I rarely do the underseat only trips now, but when I did I managed fine for up to four or even five days with the underseat bag. It depends on the reason for the journey I suppose, if hen party or other fun trip a checked bag or overhead bag is probably needed. For a sightseeing/family visit etc. trip, the underseat would do.