Getting a TCAS climb RA on an emergency descent would be an absolute s*** show. A mid air collision catastrophic. Best avoided in my book. Once both we and the pax have oxygen I try to slow the pace down. Over the Atlantic I would initiate a turn off track before initiating the descent. In busy airspace I would have a good look on TCAS before barrelling down - and think in Europe an ATC Mayday call / 7700
would be fairly high on my priority list.