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/