Vets UK do not start to assess your preserved pension claim until your birthday. They will pay the first pension within the following 30 days. They also have 30 days to pay the lump sum but, typically, they manage pay that within about 10 working days.
I have had one or two cases on the Veterans Gateway where these timescales have not been met. If you haven't got your pension and lump sum within that 30 calendar day period mentioned above, I would suggest a letter to Vets UK hastening payment and, yes, make sure it is sent recorded delivery.