How can the flight be classed as private though? Surely if it's a service offered through SkyUber an AOC would be required?
The holding out provisions in the UK ANO are a bit circular:
Art 14 Offering commercial transport and public transport flights

(1) No person may hold anyone out (whether the person who is being held out is the same person as the one who is holding out or is another person) as being one who may offer flights in an aircraft registered in the United Kingdom for the purpose of public transport or commercial air transport unless the person being held out holds:
(a) in the case of a commercial air transport operation, a valid Part-CAT air operator certificate;
(b) in the case of a public transport flight, a valid national air operator’s certificate or a valid Part-CAT air operator certificate; or
(c) in the case of an A to A commercial air transport aeroplane operation, a valid Part-CAT air operator certificate or a valid EU-OPS air operator certificate.
So you can't claim to be a CAT operator unless you have an AOC. But it does not appear to prohibit offering private flights involving passenger cost sharing, and does not in fact even require the operator to include in the offer any warning that it is not an AOC holder.