Regardless of the circumstances of the referendum, the (legal) basis for Brexit is the European Union (Notification of Withdrawal) Act 2017. I will add it in its full form so we can discuss all of the intricate details:
An Act to confer power on the Prime Minister to notify, under Article 50(2) of the Treaty on European Union, the United Kingdom’s intention to withdraw from the EU.
[16th March 2017]
Be it enacted by the Queen’s most Excellent Majesty, by and with the advice and consent of the Lords Spiritual and Temporal, and Commons, in this present Parliament assembled, and by the authority of the same, as follows:—
Power to notify withdrawal from the EU
(1) The Prime Minister may notify, under Article 50(2) of the Treaty on European Union, the United Kingdom’s intention to withdraw from the EU.
(2) This section has effect despite any provision made by or under the European Communities Act 1972 or any other enactment.
This Act may be cited as the European Union (Notification of Withdrawal) Act 2017.
That's all folks! Not much to argue about, isn't it? No "Because of the results of the referendum" and much less "Assuming that the referendum was conducted according to law". No preconditions to argue about if the Act allows for Brexit or if it does not.
Also, nothing in the law organizing the referendum stated that the results were to be automatically implemented. And even if that was the case, since the UK system works with Parliamentary supremacy1, that new Act alone would be enough to legalize Brexit, regardless of what the referendum law said. The fact that an Act (or any other law) is based on wrong or even false premises does not invalidate it legally2.
Of course, if some new information appeared that substantially weakened the pro-Brexit position (be it that the referendum results were directly false -ie. "Remain" won but the results were manipulated, or that science proves that the UK will sink under the sea the day after Brexit), at the political level it would be possible for Parliament3 to change its stance. Which would lead to the additional issue (see here and here) that the EU does not have a stablished procedure for cancelling the invocation of Article 50, so that would have to be worked out, too.
It is complicated (Politically)
Not all cheating is equivalent. More to the point, the current alegations mean that even if campaign rules were broken, people did vote freely and the results match the votes that were cast. This makes any claim for invalidating/ignoring the results very difficult and almost pointless, because:
it is difficult to impossible to prove the relationship between the rule being broken and the result (i.e., would the result have been the same wihtout cheating?).
people who became convinced to vote one way or the other are unlikely to change their vote because of financing rules violations; if some piece of info/propaganda convinced me that the best option is X, the fact that such piece was "over-budget" does not affect my conclusion. Compare that with other possibilities like ballot stuffing or buying/threatening voters.
And of course, automatically invalidating referendums for any violation of the rules is problematic, too:
Just get some underlyings to break some law while supporting the opposing side and you are set: if the referendum goes your way you won; if it does not then you are able to invalidate it.
If the results change, the people on the losing side are going to be very pissed, and even some people in the winning people are going to be worried about the whole issue.
Last but not least, a (posible) repetition of a referendum would be slow and expensive.
which advises to accept the results of a referendum (or any other votation) unless you have more than enough evidence to support that the violation of the rules did alter the results.
You might remember that shortly after the referendum there was a legal battle because the Government claimed that it had authority to initiate Brexit by itself, but it was ruled that since it affected many British laws the authority of Parliament was needed.
2 In other countries, maybe a false premise could lead to an opportunity for a law to be invalidated on grounds of conflict with the constitution or other higher level laws, but in the UK the constitution is not written and emanates (among others) from the Acts of Parliament, there are no "higher laws" that could be used to invalidate an Act.
3 Maybe it would be only necessary for the Government, since the Act does not mandate the Brexit process but only allows it. But legal details aside, it would be suicidal for any Government to cancel Brexit without Parliament's support.