As people have said, we'd just have the BNP voted in during the next election.
What people have said this? Are you really suggesting that BNP have enough candidates to stand in every constituency and that a majority of constituencies would give them a majority vote at the next general election, because this is the only way such a scenario could happen? This is real scaremongering stuff.
I would anticipate a small rise in support for some right wing parties, which would mostly come from the extreme end of Brexit voters (the EDL, NF, BNP, Identitarians etc). Should we really be comprehensively damaging our country in order to appease people like this?
As for the other Leave voters, some will by now have seen what a shitshow this is turning out to be and be glad if it doesn't go ahead, and some will rigidly stuck to their guns and be disappointed and angry, but probably not riotous.
The government should now act in the best interests of the country by cancelling Brexit and addressing the real ills in this country, such as social inequality, as a matter of urgency.