Problems with getting home assessment

My husband received the forms for migration from DLA to PIP earlier this year; I filled them out (using your guides) and sent them back to CAPITA along with all the supporting evidence I could muster. Last Friday (9th March) we received a letter giving us an appointment date for the 20th March.

My husband is totally unable to travel there by himself, or even with a companion due to severe physical and mental health problems. I rang and explained this, and requested a home visit. 'No' I was told, the health assessor had 'locked' the file, so that no home visit would be granted. I asked why and was told they didn't know.
I explained there was no way my husband could travel, and he had to have a home visit so what was the best way to get one.
I was told he needed supporting evidence. I explained I had submitted this and was then told it wasn't 'recent' enough. I was advised to get a letter from our GP but it had to be returned to them no later than the 19th March and had to be sent by signed for delivery.

That same day I rang our GP surgery, and went in with a letter for them requesting a supporting letter for a home visit (it cost £20). I set out all my husband's various diagnosis in order to try and speed things up and told them I needed it by 18th March (only that's a Sunday, so actually I would need it this coming Friday).

Today I phoned up and was told the letter might not be ready by then as the doctors are 'snowed under'. What do I do if the letter doesn't come in time please?
I'm at my wits end.

Thank you. It's the first appointment, so if I hear nothing from the GP nearer the end of the week, I'll try and get them to reschedule it. Thank you for the advice about the letter being unambiguous too - sometimes our GPs rise to the occasion, other times they are really awkward. That was why I gave them as much information as possible in my letter to ask for their support. I'll let you know how it goes...

An update...
Obtained a very good letter from the GP (eventually) which I sent to Capita. It was signed for on the 23rd April. They have only 'raised the request' yesterday (after I kept chasing with phone calls) and my husband's appointment is on Wednesday the 4th. I am now seriously worried as there has been no decision made about a home visit. He simply cannot attend and the Gp's letter makes this clear. Nobody I speak to in Capita seems to have the faintest idea what is happening.
Will they just close his claim even though we have done everything asked of us and they haven't made a decision yet? And if they do, can I appeal this with the DWP?

Thank you Gordon. I'll do that. Very grateful for your quick response.