Call me paranoid, but to my mind if you're going to try and avoid detection then do it properly. Apart from getting hold of your physical phone, you can be tracked both by your phone's sim (phone number) and its IMEI (identification) number.
First off but a cheap unlocked phone for cash, ideally at duty free or at a shop where you are not required to register it. If asked I tell them that I'm buying a holiday phone so I can use a local phone abroad. Don't pay with a debit or credit card as that will be linked to the phone's IMEI number.
I use the Vodafone pay as you go card. You can currently buy a £10 bundle for £5 and again buy for cash for the same reasons as above. It also works fine in Europe
Once registered and activated, you can reduce the monthly bundle to £5 per month which i find is more than enough to cover my punting calls and data.
They don't make it obvious but you can then top up by buying a Vodafone top up voucher from your local news agent. They come in £10 increments. Buy one for cash and you can enter the voucher number either from the phone ap or through the Vodafone website.
The system tops up your bundle monthly and debits the stored value from your voucher credits.
My experience is that many countries require all SIM's to be registered against a passport and not all the UK service providers will let you top up without using a credit / debit card - so enjoy the anonymity whilst you can