"if the individual completed their time in prison for the crime, is this legally known as a "spent" conviction?"
No, you mustn't conflate "completing time in prison" with a "spent" conviction.
The Rehabilitation of Offenderes Act 1974 provides that, in various circumstances, a conviction can be considered "spent". This only happens after some period of time (which varies by the type of conviction) elapses from the conviction. It does not follow that a conviction is spent when time in prison is served. Conversely, not all punishments are prison sentences, and it also doesn't follow that a punishment must be "completed" in order for the conviction to be spent. Consider e.g. the effect of Section 1(2)(a) on an unpaid fine:
A person shall not become a rehabilitated protected person for the
purposes of this Act in respect of a conviction unless he has served
or otherwise undergone or complied with any sentence imposed on him in
respect of that conviction; but the following shall not, by virtue of
this subsection, prevent a person from becoming a rehabilitated
protected person for those purposes
(a)failure to pay a fine or other sum adjudged to be paid by or
imposed on a conviction [...]
To summarise, a punishment can be completed without the conviction being spent, and a conviction can be spent without the punishment being completed.
This means that your criteria for declaring convictions, as currently worded, is problematic:
"if an individual has finished the punishment for their crime then that's ok as long as they declare it."
This is fine so long as the conviction is not spent. Essentially in that scenario what you are saying is that although the law does not consider the conviction to be spent, you will offer a more relaxed position and treat the person as if it were spent so long as they declare it.
However, if the conviction is spent, then Section 4(1) provides that the effect is as if the person was never convicted at all:
[...] a person who has become a rehabilitated] protected person for
the purposes of this Act in respect of a conviction shall be treated
for all purposes in law as a person who has not committed or been
charged with or prosecuted for or convicted of or sentenced for the
offence or offences which were the subject of that conviction [...]
In such a case, you cannot require the person to declare it. This is expressly provided for in Section 4(2):
[...] where a question seeking information with respect to a person’s
previous convictions, offences, conduct or circumstances is put to him
or to any other person otherwise than in proceedings before a judicial
(a) the question shall be treated as not relating to spent convictions
or to any circumstances ancillary to spent convictions, and the answer
thereto may be framed accordingly; and
(b) the person questioned shall not be subjected to any liability or
otherwise prejudiced in law by reason of any failure to acknowledge or
disclose a spent conviction or any circumstances ancillary to a spent
conviction in his answer to the question.
"if an individual has not finished the punishment for their crime, they should declare it."
This is problematic for the same reasons above. If the conviction is spent, then the person cannot be compelled to disclose it even if they have not finished the punishment.
Note that the Rehabilitation of Offenders Act 1974 (Exceptions) Order 1975 has various exceptions to the rule that spent convictions don't need to be declared e.g. for certain professions such as doctors, solicitors, accountants, teachers, etc. You should check whether or not these are applicable to your scenario. If an exception applies, note that after the passage of time the person may still acquire the right to non-disclosure, but I won't get into details of that here.
You may find it useful to read the government's guidance on the Rahabilitation of Offenders Act 1974.