It will come down to lenders discretion and criteria.

The good news I suppose is that if your lender does not accept it the very fact that you have been accepted by one lender in principle propbably means that you could (if needed) be accepted by another but before you apply to any other lender make sure your broker has verified 100% that they will accept the deposit your are being given. Make sure the broker calls the lender and discusses it in fine detail (which I am sure he will anyway) to ensure that the lender understands and is happy to accept the deposit. I have had cases where it has not been fully disclosed to the lender and then they have refused it for one reason or another.

If they are private and not open to the mainstream borrower then it is entirely up to them if they allow it or not. They are possibly not even registered as a mortgage lender and as such they will not be regulated and you may not have protection under the Financial Services Authority or Consumer Credit Act.

Unless there is some specific reason why you have chosen this lender then I would tread veeeeery carefully.

What is the name of this "private lender"

Some lenders are very dodgy about vendor gifted deposits but most are still ok.

It will depend on who you have the DIP with as to whether they will accept it. I would assume that this is a new build house and it is the builder gifting the deposit. Normally in my experience lenders are happy enough with this sort of thing as long as the deposit is no more than 5% and does not exceed any deposit value that you are placing down. As it is matching your own deposit I would think the lender will be happy about this but you will need to check with the lender and you will need to declare that you are receiving the deposit from the vendor.