Slightly more than your budget but we got this cot in the mamas and papas sale, we put it up today and it’s really good quality: https://www.mamasandpapas.com/products/solo-cotbed-cashmere-cbsoa0100 - it’s also adjustable and also turns into a toddler bed later on. For a pram, we got this which also comes with a car seat so saves money there too: https://www.babyplanetonline.co.uk/sale/mamas-papas-ocarro-pushchair-7-piece-maxi-cosi-cabriofix-i-size-car-seat-base-bundle-grey-mist?gad_source=1&gbraid=0AAAAADFqN3675UHuAc7lD7Q59efve6eoz
I would really encourage you to purchase second hand. The only things that are an absolute no second hand too me are mattress for sleep space and a carseat. Literally everything else can be second hand
Try the nursery store they do complete travel systems with car seats for under £500 x
I got the ikkle bubba travel system for £499 on sale and it's got like 12 things in it and is stunning
I got my cot from babyworld, obaby cot was £200 plus £25 for mattress. It goes lower, and can be used as a bed later so saves money. You can definitely get a pram cheaper, especially if you dont need a car seat. You can look for ones that recline fully as you don’t necessarily need a bassinet, which may save you even more money
https://www.thenurserystore.com/p/puggle-urban-terrain-sprint-gt-pushchair-storm-black?gad_source=1&gbraid=0AAAAADnW_G-EUx95tb29E0XLzac5VqQZd If you arent bothered about forward facing at the start this looks like a good option for long term, but theres lots of xheap ones on here 😊
https://www.venicci.co.uk/product/insevio-black-pearl?utm_source=facebook&utm_medium=social&utm_campaign=outlet_cold&utm_content=ad1-outlet-grafika-1
https://www.babyplanetonline.co.uk/tutti-bambini-malmo-cot-bed-with-cot-top-changer-mattress-oak?gad_source=1&gbraid=0AAAAADFqN34qXDNiwJfbHo7z8yx_AJf78&gclid=Cj0KCQiAwtu9BhC8ARIsAI9JHanGlvCoy-LLuhFi_0BJglLqBU9Qk7MR9lRDTJ2WXSp-6mEapapAfUsaAjUAEALw_wcB I managed to get this cotbed on Robert Dyas for £169 not sure if you shop around you’ll find it cheaper somewhere
Try babyworld or pram world, all main brands but discounted we saved 1,000 on our pram using the sites