The fastest way to get Bitcoin instantly without ID is usually physical Bitcoin ATMs, available in most cities.
The disadvantage is that often fees are high, rates not very good.
Where I live the next ATM is a 3minute walk from my home. I can get BTC this way within 20 minutes without ID. 24hours a day, 7days a week.
For UK see for example here:
https://coinatmradar.com/country/225/bitcoin-atm-united-kingdom/You're lucky. I need to take a bus to get to my nearest ATM. I have been using the ATMs exclusively to turn my £fiat into BTC.