Personally I don't think anyone outside the divorcers give a flying fuck what reasons are cited on divorce papers. My exh cited awful things when he petitioned and i didn't contest so it was over quickly. No one ever asked me what was cited!
And if I had the money to pay costs, I'd do so just to get it over with asap. I won't let misplaced pride stop me from a fast and relatively painless divorce. Let her think she's 'won', as long as you get what you want (presuming you do want a divorce asap?).
Certainly you really need to get specialist legal advise before ignoring the papers! That could completely backfire if you do not know the legal standing of Cypriot divorce jurisdiction in the UK! Would the costs rise, would you then be automatically responsible for them?
Taking care of the dc and paying everything for them will not have an affect on the divorce petition. She is morally in the wrong obviously, but you need to focus on the legal. Divorce courts don't care she's a shit mum and a shit person; they only care about the law.
If you ignore the papers or contest the reason and costs, might she retaliate by making a claim on your finances? Or drag the divorce out which could cost you a lot more over time? Is she the kind of person who will make things more difficult as 'revenge'? Demand access to the dc pr even residence, for example?
If divorce is your goal, don't let pride and moral right stand in your way. Don't stick your head in the sand - get proper legal advise. Do what is best for you and your dc, as you have obviously been doing so far.