What does 'regaining control of our borders' in Brexit campaigns actually mean for non-EU immigrants to the UK? Does it have any bearing at all on this group of people?
Actually, it does not affect non-EU immigrants at all.
EU laws grant that nationals of EU countries have the right of residence in other EU countries. This is not a "free for all" right, though; after a few months of residence they have to show they have a way to sustain themselves (v.g. a work) or may be deported.
People who are not national of EU countries (the usual expression is third country national) are covered by the Schengen agreement. The Schengen Treaty led to the abolition of internal borders between the signatories. When a third country national is granted a Schengen visa (by any country in the agreement), the visa is 1 valid to visit all of the Schengen countries (although you are supposed to apply to the country at which you will be staying more). In exchange, there is a common policy for issuing such visas2 and the information of applicants is shared between the Schengen states3.
The question is that the UK (and Ireland) got an opt-out of the Schengen agreements, that allowed them to be part of the EU and not of the Schengen Treaty. So in the UK, for third party nationals, visas issued under the Schengen Treaty are not valid.
So, the rules for allowing third party nationals inside the UK are already under control of the UK government, either by specifying for each third party national which ones are free to enter without visa, which ones can enter with certain visas4, and which are the requirements to get an UK issued visa.
A couple of links:
What does European Union "freedom of movement" mean, and how much can individual countries restrict intra-EU immigration? (this is about the EU nationals).
1 Or should be, as some national legislations fail to fully comply with the Schengen agreement.
2 Due to the refugee crisis there has been some talks to create just a single entity for issuing Schengen visas, but at the time of writting there is not a solid initiative towards that.
3 IIRC some states might still issue "national" visas that are valid for their territory but not for any other country; but I am not fully sure about that.
4 As Bregalad points, there is at least one situation in which the UK allows entry with a visa issued by another country; it looks like Indian and Chinese nationals may enter the UK using some visas issues by Ireland (and viceversa). However, the treaty governing this is not related at all to the Schengen Treaty or to the UK being a member of the EU.
Contrary to the other answers/comments here, the actual answer to this question is that it makes a real difference.
Brexit means that the laws governing immigration (from EU and from non-EU states) will be made solely in London, whereas for the past 40 years they have been made mainly in Brussels.
"Control of our borders" includes the repatriation of the legal powers to control all entry into the UK, i.e. entry from all other countries.
Non-EU immigrants can obtain rights in other EU states, i.e. other than the UK, and then are entitled to enter the UK as EU citizens, under EU freedom of movement rules. This may take 5 years of settlement in, say, Germany -- but this became a significant problem once millions of refugees a year began being accepted into Germany.
It is not enough to say 'it can't happen because of Schengen'. Once a migrant has acquired German citizenship in short order, and become a "German", he/she/they can not then be refused entry into the UK if we are still afflicted with the freedom of movement rules that force us to accept entry by all EU citizens.
People who are not national of EU countries : This is certainly the nub of it, because this qualification ceases to have effect once the German government grants German nationality to the millions of immigrants queuing up to claim it.