What is the needed budget to start a succesful private brand like allbirds or happyskinco or snow teeth whitening.?

That's a huge question that can only be answered with the "depends" word! But a good way to start is to look at the market. Doing a market analysis to give you an idea about the demand and competitors would be a good way to start.
Then look at your production costs - is this a product you can license, resell, or do you need to create from scratch?
Then, of course, the administrative, setup and foundational needs will be front-loaded, so be prepared.
That's where I would recommend you begin, and that will help to start framing the idea of whether you are talking about $1MM, $100k, or $1k...
-JL

Allbirds is pretty straight forward if you do your online research, this article is a casestudy of allbirds, https://www.privy.com/blog/allbirds, you will see that he started out raising money through a kickstarter $119K and 2 years later after testing and lab work he launched with a marketing budget of $400K. Brand Strategy alone can run $50K and then you have to add on identity design, product design packaging, and online sales. All of this creates the full brand plan from strategy, design, production and website for $100K package then you would pay additional for a Marketing/Social and Sales launch.

As for happyskinco, they are a shopify site and product development brand that would start with Brand & Product Strategy work, I would say you would want at least a $60-80K budget to launch a similar brand and I would focus on Kurt Elster to help me develop the shopify site -https://ethercycle.com/

Hope this helps you to see the two brands you referenced have a well developed brand and it took allbirds since 2014 to grow the brand and presence.

Branding is a process NOT a single event.

A private label business continues to sound more and more like a lucrative venture as time goes by. Private labelling can indeed be a gold mine – if you know what you are doing. The truth is times are changing and the money you would have needed to establish a private label brand a few years ago now will not be enough. To succeed in a private labelling business, one of the most important things is an adequate capital base. For this to be attainable, you need enough money to secure a large inventory of your product, create a good brand and market it well. The reality is that not all qualified manufacturers or suppliers choose to list themselves in B2B sourcing platforms or directories, another option to find a reliable private label manufacturer is to hire a reputable sourcing agent or work on joint order with the sourcing departments of some worldwide retailers. You can therefore focus on other aspects of your business while starting out and enjoy Prime Eligibility of your product. Also, you have no inventory to call upon when there is a product shortage from the supplier, which can be very risky. This is where you are in full control of the handling of your product. You take care everything from inventory and packaging to shipping the product to the customer. With this control, you are always aware of what you have at hand, so you can avoid making false promises to customers.
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath