Sorry OP, but you don't seem to understand. A Will and an Executor are only necessary when someone DIES.
NO ONE should be distributing your Father's money, he IS NOT DEAD!!
As your Dad didn't have the foresight to get a POA done while he was still mentally stable, no one can act on his behalf until they have been appointed Deputy by the Court of Protection. As you've already found, this is a lot more expensive than had he done a P.O.A., but that ship has sailed now. Getting the deputyship, may also take some time.
Meanwhile, who on earth thought it was a good idea to sell your parent's house, and has the sale actually gone through? If not, then put a stop to it, as now your father has no capacity, assuming he was either the sole or a joint owner with your DM, then he cannot sign to sell the property.
Once you have Deputyship, then whoever has that, which should, unless your parents were planning on separating, be your Mum, then she can act on behalf of herself and your Dad. The fact that your Mum has appointed your sister? as her POA, does NOT necessarily mean that your sister can spend any of your Mum's money, unless your Mum gives her permission to do so. It will depend on what she has put on the P.O.A. She may have agreed that your sister, can do certain things for her, which I believe would have to be stipulated, although I'm not 100% sure on that, or she may have stated that the P.O.A. is only actionable, as and when she loses capacity to make her own decisions. You need to talk to your Mum and see if she will let you have a copy of the P.O.A. so that you can see what it says.
Also, has she done a P.O.A to cover her health? This is a separate document. As if not, she really should do, so that should anything go wrong, and she be unable to make a decision after, for example a road accident, or an operation, you and your siblings will not have the right to make any decisions about her, for example having machines turned off, etc.
This sort of post is something that should make us all think. Whether we be older parents ourselves, or the children of older parents, with youngsters of our own. It really is INCREDIBLY important to have P.O.A.s for both finances, and health in place BEFORE anything goes wrong, dementia, accident, etc.