Does the Funeral Director need a death certificate (MCCD)
before the body can be removed from the place of death?

Ideally the doctor should view the body and issue the certificates first.
This can be a problem if the doctor is not readily available
- see Law Commission 2015 report - section 3.27 to 3.30, pages 54-55.

Section 3.30 of the Report: "we consider that the law currently
not prevent removing the deceased body, or even embalming it
prior to obtaining the MCCD.

Thus a funeral director can reasonably remove the body and embalm it without the MCCD.
The doctor should first be contacted to confirm that he or she will issue a certificate
- does the doctor need to view the body before removal or refer the case to the Coroner? 

Two aspects can be confused:

1. "Transfer" of the body (e.g. from where the patient died to the funeral director). 
B&C Act section 46F:
- (1): a MCCD is required before “charge of" a bodyis transferred to another person
- (2) specifies that funeral directors are exempted from this requirement. 

2. “Disposal" of a body refers to burial or cremation;
- it does require a MCCD (B&C Act section 46E).

Burial & Cremation Act 1964 s46F: Transfer of charge of body (excerpts):
(1) A person having charge of a body must not transfer charge of it to another person
unless he or she first—
(a) gives the other person a doctor's certificate or coroner's authorisation; and
(b) gets from the other person a notice, in the standard form, taking responsibility
for notifying a Registrar under section 42 of the Births, Deaths, Marriages, and Relationships
Registration Act 1995 of the death of the person whose body it is …

(2) However, in any of the following circumstances the person transferring the body
does not have to comply with subsection (1):
(a) a person having charge of a body who is not a funeral director transferring charge
of it to a funeral director:
(b) a person having charge of a body transferring charge of it to—
(i) a constable; or
(ii) a doctor who has been directed by a coroner to perform a post-mortem …

(3) Nevertheless, if a person has a doctor's certificate or coroner's authorisation at the time
that he or she transfers charge of a body to another person,
he or she must give the certificate or authorisation to the other person.

Burial & Cremation Act 1964 s46F. Bodies to be disposed of within reasonable time
(1) A person having charge of a body must, within a reasonable time of taking charge of it,—
(a) dispose of it; or
(b) cause it to be disposed of; or
(c) transfer charge of it to another person for—
(i) disposal; or …