To understand questions such as these, we need to distinguish between spiritual fact and practical expression. The fact is that wherever there are believers in Christ, the church is there. The biggest shortcoming today with regard to the church is not the fact; the gospel has been preached throughout the earth. But a practical expression of the church according to the New Testament pattern is hard to find.
So we do not claim to have gatherings of the whole church of God in London. The believers in London meet in many different places, mostly not with us. The mark of those who belong to the church in London is having been saved by faith in Christ. Even in regard to the gatherings of believers, no group has the basis to claim that God is only working through them. In whatever way and to whatever extent God's people touch something of His desire, He will bless them. If God waited to bless until His people matched His desire in every respect, He could bless no one. It follows that God's blessing does not mean that we have arrived. There is always more for us to gain of the Lord and to cooperate with Him for the accomplishment of His purpose.
Having said all that, it is still important for us to seek the Lord's will for the practical expression of the church. The Lord desires that the church would be a testimony of Himself both to the world (John 17:21) and to the fallen angels (Eph. 3:10; cf 6:12). The world will not be impressed by a "spiritual" oneness among believers scattered among many different Christian groups. While all the believers in London belong to the church in London, not all are in a practical oneness. Some need to put aside all divisive things in order to put into practice a practical oneness that the world can see.
So the question becomes, what are the things that often break the oneness among God's people. The most obvious is the names by which we identify ourselves. Many believers feel that there is so much taught in the name of Christ that they cannot agree with that they need to distinguish between themselves and other Christians. The name identifies the Christians who are the same as they are in teaching and practice. However, that is the problem with names other than Christ. God's people are not all the same in teaching and practice. There are some basic truths that distinguish between believers in Christ and the people of the world. These cannot be compromised. But there are many other things in the Bible to which we should apply the principle in Romans 14 of receiving one another without attempting to make others the same as ourselves. Names other than Christ or Christian automatically exclude many of God's children and any oneness among that group of believers is less than the oneness the Lord desires.
The matter of different teachings is very difficult to reconcile with oneness. In general terms we can say that we should not accept anything that is against the Bible and not insist on anything that is not in the Bible but Christians can find plenty within the Bible to condemn one another about. There is a central line, a basic thought, within the New Testament that should guide our understanding and application of the whole Bible. There are many other things in the Bible that, even though they are proper and helpful, are less crucial and to which the principle of Romans 14 applies. To exalt any of the minor things and insist on it is also divisive.
The same can be said about ministers of the Word. The Corinthian believers received help from several ministers (1 Cor. 1:10-13; 3:1-9). Their problem was that they paid too much attention to the ministers rather than the life and light supplied through their ministries. As a result, they each had a preference for a certain minister and were making distinctions among themselves based on their preferences. That does not mean that the ministers had the same portion. For instance, Paul's ministry was, no doubt, different than that of Peter, and deeper, richer, and higher than that of Apollos. But to have a special fellowship with those who appreciated the same minister was fleshly and indicated their immaturity.
It is interesting to note that the ones who declared, "I am of Christ" (1 Cor. 1:12) were also being fleshly and divisive just as the others were. It is absolutely correct for a believer to say, "I am of Christ." The problem was that the ones saying it in Corinth were using that truth, concerning which they were clearer than the others, to make a distinction between themselves and the others. All the believers are of Christ, not just the ones who proclaim it. We Christians can, and often do, distinguish ourselves from others based on the truths we have seen and appreciate. If we allow the Lord to work the truth into us so that it becomes practical and real in our daily living, then we will be able to minister it as a life supply to others. If it is only doctrine to us, then we can only debate and condemn others who don't see it the way we do.
Insisting on or forbidding certain practices is another thing that breaks the oneness of believers. Of course, the church cannot accept sinful things but Christians often cannot agree even on scriptural practices such as baptism or speaking in tongues. If we are to be practically one with all of God's children then we cannot insist that others conform to our practices. The Lord's receiving is not based on practices; ours cannot be either. If everyone in a fellowship of believers is the same in practice, then that fellowship has no place for others who are different and is exclusive.
The oneness that the Lord Jesus enjoys with the Father, that He also desires for His people (John 17:22), is of the Spirit (Eph. 4:3) and in the spirit. If we are not living in the spirit then even a harmonious situation is short of the Lord's desire. When we are enjoying the Spirit, all of God's children are lovely to us; we have no problem keeping the oneness of the Spirit. When we are not in spirit, we are already outside the realm of the oneness of the Spirit even before any issues arise between ourselves and others. If we are right in every respect but miss the Spirit, we miss everything. May the Lord save us from ourselves and our own opinions to the extent that we can be one with all His children. This is the mark of a proper practice of the church life.