Can I reclaim VAT on costs from before I registered?

Good news — your understanding is exactly right. You can reclaim VAT on goods bought up to 4 years before registration, as long as you still have them (so your laptop and any unsold stock qualify), and on services bought up to 6 months before registration. The purchases must relate to your business's taxable supplies, which they do here. You don't need to do anything special on your first return — just include the VAT in the reclaimable input VAT box as normal, but do keep the original invoices as evidence. One thing to watch: if any of those goods have since been sold or consumed, you can't reclaim the VAT on them.