The builder is likely to be in breach of the terms of the lease by not doing their job. What this means for you is you can involved the First Tier (Property) Tribunal to remove this management company and to install a new one, to ensure good management of the property going forwards.
Given the complexity you may well need a law firm to help you with this - either of the leaseholders can pay for the application or you can share the cost.
There are a couple of specialist law firms which can help and I will mention them shortly.
The tribunal route is quicker than the county court claims procedure and after receiving the application, the tribunal will then make a decision and indeed it has the power to remove a management company for another one to ensure good management of the property.
The Tribunal is made up of 3 people: A lawyer (Chairman), a surveyor/valuer (who could also be Chairman), and a lay person (member of the public). You / your lawyer can present the dispute to them. The following would be able to do this for you : https://www.bishopslaw.co.uk/first-tier-tribunal-property-chamber-advice/ or www.bradysolicitors.com who have specialist departments for these Tribunal claims.
The latter firm also offers a free guide on how to change managing companies : https://www.bradysolicitors.com/taking-back-control/changing-managing-agent/
The hearing itself is quite informal. Once made, the order is then binding - if the order is ignored then the county court will recognise it and you can apply for an injunction to force them to comply.
After receiving the application, the Tribunal should then make a decision within 6-8 weeks. So not long, certainly not compared to a court application which could take several months.