Magistrates' Courts only deal with criminal cases. (OK, there are a few exceptions to that statement but they're not relevant here). If the matter goes to court, it will be a County Court that hears the case.
Cases in the County Court are decided upon a 'balance of probabilities' basis. (i.e. neither side is required to prove their case beyond doubt. The judge simply has to decide which side's version of events is most likely to be true).
It's not clear from your post whether it's the driver of the car that hit yours who's seeking to initiate court action against you or the driver of the car that your vehicle was then pushed into. From your account of the accident though, it's hard to see how either of them could convince a court that they (and/or their passengers, fictitious or otherwise) suffered injury through any form of negligence on your part. So I find it hard to see how they could win their case.
If the judge ruled against them though, it wouldn't actually be proof that they were lying; it would simply mean that he thought that your version of events was most likely to be the true one. So there could be no 'punishment' for lying involved.
It's only in extremely rare situations, where it can be proved that a person who initiated court action was lying, that 'punishment' could come into play. However that punishment wouldn't come from the County Court judge; it would require a successful criminal prosecution for perjury and/or for 'attempting to pervert the course of justice' in order for such punishment to be meted out.
For example, Jeffrey Archer launched, and won, a libel case against the Daily Star. However he was later proved to have lied in that court case, resulting in him being convicted of both perjury and perverting the course of justice and sentenced to four years' imprisonment.
As I've said though, such cases are extremely rare. Vast numbers of people lie in court every day but hardly any are ever prosecuted for it.