Grammatically, 1 Corinthians 13:9 is ambiguous enough that Paul could be referring to the parts of the body of Christ ("individually"). However, the context makes it clear that this is not the case and that he has the more common meaning of "partially" in mind.
In chapter 12 of 1 Corinthians, Paul first lists a number of possible spiritual gifts and uses a "one body, many parts" analogy to explain why different believers receive different gifts. Toward the end of the section, Paul says:
ὑμεῖς δέ ἐστε σῶμα Χριστοῦ καὶ μέλη ἐκ μέρους. (12:27, Westcott & Hort)
Now you are the body of Christ and individually (ἐκ μέρους) members of it. (ESV)
Here, the meaning "individually" is clear, as is makes little sense for Paul to be saying believers are only partially member of the body - surely he does not mean believers are only partially members of the church. He concludes the section by saying one should "earnestly desire the higher gifts" and says:
And I will show you a still more excellent way. (12:31)
This phrase is more logically connected to chapter 13, where Paul outlines the better way, that is, love. In verses 1-3, he states that various spiritual gifts are worthless without love. In verses 4-7, he describes what love is. And then in verse 8-12, he makes his argument as to why love is the "more excellent way". Verse 13 is the conclusion to his argument.
With this is mind, it will become obvious that ἐκ μέρους in these verses means "partially." Paul's first argument is that love is unending whereas prophecy, tongues, and knowledge will not last forever.
Love never ends. As for prophecies, they will pass away; as for tongues, they will cease; as for knowledge, it will pass away. (13:8)
Paul explains continues this argument in the next two verses:
For we know ἐκ μέρους and we prophesy ἐκ μέρους, but when the perfect comes, the ἐκ μέρους will pass away.
In verse 10, Paul is thus contrasting "the in part" with "the perfect". When the perfect comes, the imperfect ceases to exist. When the believer achieves the "higher gift" of love, the lesser gift of prophecy/tongues/knowledge ceases to be relevant. The moment when the perfect comes is not referring to an eschatological event (the return of Christ), but rather to the spiritual maturity of the believer - a theme that Paul has been discussing throughout the letter. If in verse 10, ἐκ μέρους means partial/imperfect, it is only natural that in means likewise in verse 9, which is part of the same sentence/thought.
Verse 11 continues the argument:
When I was a child, I spoke like a child, I thought like a child, I reasoned like a child. When I became a man, I gave up childish ways.
reinforcing the conclusion that Paul has the spiritual maturity of the believer in mind. Does the believer achieve maturity by leaving the body of Christ? Certainly not. Maturity is achieved when the partial gifts ("childish things") are left behind for the superior gift of love.
Paul's three-fold argument is thus:
- Love doesn't end; other gifts do
- The perfect gift comes; partial gifts are obsolete
- Maturity is reached; childish things are put aside
Verse 12 provides further evidence that Paul means "partially" here. In this verse, he draws an analogy with looking in the mirror. To understand the analogy, we must realize that in ancient times looking into a mirror provided a poor representation of the individual. Thus, there was a significant difference between an unclear image in a mirror and seeing someone's true image "face-to-face". This analogy makes perfect sense with a "partially" understanding, but makes no sense in a "individually" interpretation.
The question implied that the "partially" understanding creates a contradiction with some other Bible verses, which perhaps imply that the mature church has complete knowledge. However, this is not the case. As already noted, Paul is talking about the "immature" believer in 1 Corinthians 13:9-10 and I argue this supports a "partially" interpretation. As such, even if the other verse are saying as the OP suggests, there is no contradiction. Even so, I don't think the same idea is in mind in any of them.
First, Romans 15:14:
I myself am satisfied about you, my brothers, that you yourselves are full of goodness, filled with all knowledge and able to instruct one another.
Here Paul is addressing the Roman church as a whole ("my brothers"). Even if they are collectively filled with "all knowledge", there is no reason to equate that with meaning individual believers (which is who is in view in 1 Corinthians) are filled with all knowledge. Furthermore, it is really unlikely that Paul literally means "all knowledge of any kind", but rather he only means sufficient knowledge in spiritual matters to allow mutual correction and learning.
1 John 2:20,27:
But you have been anointed by the Holy One, and you all have knowledge. ... But the anointing that you received from him abides in you, and you have no need that anyone should teach you. But as his anointing teaches you about everything, and is true, and is no lie—just as it has taught you, abide in him.
The key to understanding what John means by "knowledge" is found in verse 19:
They went out from us, but they were not of us; for if they had been of us, they would have continued with us. But they went out, that it might become plain that they all are not of us.
Essentially, John is saying "some apparent believers left the faith (v. 19), following false Christs (v. 18), but you won't leave the faith because you have the proper knowledge (v. 20)." The knowledge he refers to is thus an element of true belief, something all believers have, not the spiritual gift that Paul is talking about.
Beloved, although I was very eager to write to you about our common salvation, I found it necessary to write appealing to you to contend for the faith that was once for all delivered to the saints.
It's not clear to me why the questioner feels this verse implies "all Christians [have] all truth". Perhaps a misunderstanding of the archaic "needful" of the King James? In any case, I can't see how this offers even an apparent contradiction with the "partially" interpretation of 1 Corinthians.