No you don't have to wait that long. You have to ask your existing provider for a PAC code and they have to give you this within 48 hours. But you do have to ask them. Are you staying with Vodafone? Ie changing from PAYG to a monthly contract? If so then you might not even need a PAC code....you just need to speak to the right person at Vodafone.
Once you have a PAC code then there is a time stipulated (24hr IIRC) unless it's a Friday in which case you have to wait until Monday as no changeovers take place over weekends.
I have just gone through similar hoops. I was with Orange for many years on a monthly contract which came up for renewal earlier this month. A couple of months before I scouted around for what was on offer. Long story short, Tesco could do me a much better deal than Orange. Orange couldn't even come close although they did try. The bizarre thing is that the network Tesco are using for this particular deal is Orange! I asked Tesco's about PAC codes and was told that if you are on a monthly contract with Orange and moving over to another monthly contract with them but via a third party eg Tesco then their systems can't support a PAC code transfer in this instance (maybe true of all mobile providers, I know not). Instead you have to convert your existing monthly Orange phone to PAYG then ask for a PAC code, give the PAC code back to Orange who will then do the number transfer to your new phone. That, of course, is the theory. What they forget to tell you is that to do this you have to deal with a call-centre in India and while I love the country and the people, often (as is the case in the UK I hasten to add) the person you are speaking to is often not the sharpest. There is also the delay between the stages when your old phone is out of action.
A very very tortuous process. I eventually got to the point of requesting a PAC code for the nth time and when asked why tried to explain. Fortunately I finally got put through to someone who simply said 'Oh, you don't need a PAC code for that, I can do it for you now'. ](*,)
One final thought. If you are happy with your existing handset then there is an even cheaper option for you which is to go SIM only. Still on monthly contract but because they aren't stumping up for a new handset you get some of the cost saving.