There is no limit as far as i know.

Here is a guide how to transfer it.
https://steamhelp.yuanyoumao.com/en/faqs/view/7EFD-3CAE-64D3-1C31#transfer

Just ask your old provider for a PAC code, give it to your new provider and you can take your number with you. everything will work like you've always had that SIM.

Note that you don't need to remove the authenticator to change phone numbers

https://proxy.yuanyoumao.com/phone/edit

The phone number is a recovery mechanism. You can change your phone number by using the Steam Authenticator itself. The authenticator doesn't change because your underlying phone number changes. I swap sims out frequently when travelling so my 'phone number' changes often.

Just go to that website, use the "yes use my authenticator" option. it sends a notification to your existing authenticator over wifi. And then change your phone number at that time.

Also note that most countries allow you to maintain your phone number when switching mobile providers. So you should really look into that option as the UK definitely allows for this. Otherwise one other common option is to move your existing phone number to a VOIP service first like Google Voice, then move it back to your new mobile provider afterwards. But your old provider should give you a PAC code to allow you to migrate your phone number as long as you keep that service live during the transition.