There’s not really a thing such as minimum length yours needs to be, a good hairdresser should be able to work with most lengths to get it to seamlessly blend. It depends what method of extensions and how long you’re after in terms of cost, extensions are normally sold between 18”” and 24”. So for example mine is the 24” weave from Remi Cachet. They do different ranges at different price points too. Mine is this one
https://www.additionallengths.co.uk/products/full-flat-weft-colour-1b-length-24 so £470 for the weave and then £60-80 to re-fit every 6 weeks. Nanos are definitely cheaper though. And I don’t know if you’d want so dramatically long if it’s currently short. Then the shampoo I buy tbh I should be going more expensive but I just use any cheap sulphate shampoo in Boots so that’s just normal price.
Just checked and the nano equivalent in my length and colour is £380. Nanos is different in the sense you buy them in bunches of 25g so it depends how many you need in your head. Mine would only be for thickness so I’d buy four packs of these. In comparison, 20” is only £280.
https://www.additionallengths.co.uk/products/nano-tips-colour-1b-length-24-inch
Ultimately the best would be to find a hairdresser to give a consultation specific to your hair and they should be able to colour match you, price it up and let you know