'Vests' are sold as bodysuits - i.e. sleeveless, short or long-sleeved t-shirts with poppers between the legs. Sleepsuits are 'baby-grows' in old money... The vests/bodysuits go over the nappy, under the sleepsuits/babygro. Usually they stop having poppers on the vests at toddler age, so you will be unlikely to find anything different to those you've bought for a newborn.
The usual advice on layers needed for a newborn is a short-sleeve vest/bodysuit & babygro plus blanket & sheet at 16ish degrees, if colder, long sleeve vest & more blanket, warmer, less blankets. I would say that, unless you are in the arctic, you will probably not need the long-sleeve bodysuits/vests at this time of year but a short-sleeved one under the babygro will suffice.
A general rule I was told was that babies need 1 more layer than you, so if you're in a long-sleeve t, they need a vest + plus long-sleeve top. Still use that rule with my DD & she's nearly 2yo!
You will need LOADS of vests/babygros - factor in massive runny poos up the back, sick, milk leaking everywhere, & you may find baby wears 4 different outfits a day!!!
I never used scratch mitts, DD just refused to keep them on. NEXT apparently do babygros with built-in scratch mitts that kind of fold-over the hand. If you have a Primark, they used to do packets of scratch mitts dead cheaply, so a few packets won't break the bank if you never use them...