I’ve got a six week old, so in the thick of it.
Wouldn’t get a Sleepyhead until you know you need one.
Bouncy chair- yes, it’s been a hit here from very early.
Playmat - not really of use yet here, I’d buy if there’s a particularly good deal or you have the storage for it.
Sling - library, as others have said. I was adamant I’d not feel comfortable with a fabric wrap, bought an ErgoBaby, you can guess what happened next.
Clothes - they wear some things for five minutes, so I’d completely buy some stuff used. I personally think outfits are a waste (mine is in sleepsuits 24/7) but it’s up to you.
Baby bath - I found invaluable, if not used then new for £16 at Mothercare. Or use your sink to begin with.
Pram/buggy - I’d buy used if possible and you find a good condition one you like. We ended up with an ex-display Cosatto for £240 instead of £500, without much trying. They are expensive new so something to buy used if you can, imo.