Help with Conversion tracking across 2 domains

I am looking for some help with Google Ads tracking. We work with a white label which we promote via our Wordpress site in Google Adwords. Recently, the white label supplier has upgraded their site and the Adwords tracking has broken. What with all of the recent Google tracking changes, the tracking code on the site is out of date and needs to be cleaned and fixed. We use Tag Manager, but the code has evolved into a mix.

I would like to track a user from a Google Ads click through to our Wordpress front end site (domain 1) and then onto the white label (domain 2). I would like to track registrations (sign ups) and FTDs (First Time Deposits), so ideally I would like someone to document the whole process and compile suggestions/instructions for the white label supplier to implement on their domain (domain 2) such as firing a pixel at the appropriate point.

Making changes on the front end site is easier as we have full control of the server here.