Having had similar issues, I’ve stopped applying for IHT numbers. I simply send in the IHT forms along with a cheque with the name of the deceased written on the back.
It is also worth noting that complaints against HMRC do speed up the process. I’m now at the point where I diarise my chasing letters and complaints ahead of time, and have template letters for this purpose. Initial letter, a chasing letter a month later, a second chasing letter a month after that, and a formal complaint a month after that. It is not unusual for me to have half a dozen open complaints against HMRC at any given time, but whatever it is I was complaining about is generally sorted within 30 days of the complaint letter. In one case I was even able to get them to pay our costs for chasing them.
Even three years ago I could not have imagined having to take this approach to HMRC, and I can only speculate as to how demoralising it is for the remaining staff.
Elliot, Bond & Banbury