Hello and welcome to SAYNOTO0870.COM.
Searching around the internet, I have found HMRC numbers in the HMRC office beginning 01634 8217xx and 8218xx, so perhaps worth trying around that area, i.e. 821700, 701 etc.
Whatever you do, post back what you find as this may provide clues to help others to find an alternative.
P.S. I called the 0845 601 2877 number and it says for VAT "Time to pay" queries press 1 and for any other Time to pay queries ring 0845 604 9051. Pressing option 1 gets to a message apologising and then says that all agents are busy and to call back later and then the call ends.