HTML site Moving Server and CMS to Wordpress

Here is the senario;

My client has an html site located on xyx server.

I have agreed to create a new wordpress website on my own servers but the main point is there must be no downtime between changing the dns settings from the xyz server to mine. This means I need to have the wordpress site set up and created with all the relevant content prior to switching the dns across to my server....

I have been creating the new WP site using the IP address temporary url 123.00.00.21/website.com but I subsequently realised the WP tables are creating urls which include the ip address. It seem the urls of the test site would need changing prior to the switch over....

I have tried a number of different methods including the above as well as installing WP on my servers and simply uploading the clients html website into the root folder - I thought I could simply change the htacccess file to point to the index.html whilst still being able to create the wordpress site (I could then switch the dns settings from the old site to the new and visitors would still see the old site until I am ready) in the same root folder - simply switch it back to index.php when I am ready for the site to go live but that seems to screw things up....

So, I am wondering whether there is a simple and straight forward way to accomplish this switch which I am missing?

Any help appreciated.

WP doesn't store IP but URL/domain of site.
As you set your IP as domain in installation you need to change in settings database that IP to real URL on two places.
Just go in phpmyadmin and look in settings table inside WP database.
Find two places where is your IP and change it to URL you want.
If you need help send me pm with login info and I will fix it in few minutes.

like deki said,.
and what i use to do, for example if i install working site at : 127.0.0.1/mysite.
after its completed, i do an sql export of whole database, do a search for "127.0.0.1/mysite" and replace that with the correct domain name.
after that dump all existing tables, import the modified sql file, and then change the DNS on the domain

Or you can simply use backupbuddy to move WP from one place to another.
It will do all changes.
I think there is nulled version here or on some other forum, but yes you can find it for free if you search around.

complete the wordpress site offline and when you are ready sign up to cloudflare

Thanks all for your comments. On a test site I recreated the same conditions (i.e. ip/domain and went to 'general settings' and deleted the reference to the IP address at the start of the url. When I visited the site after I changed the dns settings (just the url no ip) it worked - does changing the urls in general settings rewrite the tables or did I miss something there?....