You switch web design studios without losing the site by doing four things in a fixed order: secure what you own, build the new site where nobody can see it, change DNS only after it passes testing, and move the domain as a separate job. Most of the damage we hear about comes from steps happening out of order. The old studio being difficult is rarer than people fear.
Start with a look, not a request. Find out who is listed as the registrant on your domain and whether you can log into the registrar account yourself. Then find out who pays for the hosting. If either answer is "the studio, under their name", that is the first thing to fix, and it is much easier to fix before anyone has been told you are leaving. Pick the new studio first and let them run the handover requests. They've done it before, and the old studio is more likely to answer a professional than a customer who is cross.
Next, what has to be in your hands. The registrar login, in your business's name. The hosting login, or a full copy of the files and the database. An administrator user in your own name on the CMS, whether that is WordPress, Shopify or Webflow. Admin access to analytics and Search Console. A written record of every DNS record on the domain. A backup and a content export. And a list of every paid plugin, theme, booking tool and payment tool, with who pays for each and when it renews. Source code is a separate conversation. Ask for it by name.
Then the order. The new site gets built on fresh hosting and tested at a temporary address. DNS changes only when testing passes, and the old hosting stays live for about two weeks afterwards in case something turns up. The domain moves separately. A domain transfer does not take the site offline, provided nobody edits DNS while it is happening. For a .uk domain the new registrar supplies a tag and the current one updates it, usually the same day. For most other endings you release the transfer lock, request an authorisation code and hand it to the new registrar, which tends to take five to seven days. A domain registered or transferred in the last 60 days may be locked, and changing the registrant details just before a transfer can trigger the same lock, so leave those edits until afterwards. Rules in this area have been under review, so check the current position with your registrar before you plan around the 60 days.
Email gets its own paragraph because people forget it until Monday morning. Mail is routed by MX records. If your mailboxes are hosted separately, on Google Workspace or Microsoft 365 say, copy those records across exactly as they are. If the mailboxes sit on the old hosting, the mail move needs its own plan, and don't leave it to cutover day.
Search traffic is the part that costs real money when it goes wrong. If the site keeps the same URLs on a new host, Google has nothing new to learn. If the design or the platform changes, every old address needs a permanent 301 redirect to its new equivalent, page by page. Before the switch, export your Search Console data and note your best-performing pages, so you know what to check. Google's Change of Address tool says to keep redirects in place for at least 180 days, and its site move guide says generally at least a year. We'd keep them a year. After launch, watch Search Console for crawl errors for several weeks.
If the old studio stalls, ask in writing for the domain, the site files and the access, with a reasonable deadline, and let the new studio handle the correspondence. For a .uk domain where the provider has gone quiet, Nominet has a process for registrants who can prove ownership, and invoices naming your company help. Meanwhile the new studio can build the replacement in parallel, ready to go live the day access arrives.
Before you sign with the next studio, get three things in writing: the domain, the hosting and the accounts are in your name, who your day-to-day contact is, and what support you get after launch. Then ask them to walk you through the order above before the first invoice goes out. A studio that can do that calmly has moved sites before.