One way to test polyploidy is probably to realign raw reads against the assembly and counting read frequencies of variants.
In a diploid genome, we expect to find predominantly variants supported by 50% of the reads (two alleles, heterozygosity). In a triploid genome, we assume to find variants supported by 33% and 66% of reads (three alleles). Tetraploid: 25% and 75%. And so on.
I expect many caveats of this approach though. First, it assumes that all polyploid chromosomes are truly collapsed into a single chromosome in your assembly and not assembled as separate contigs. Second, it requires very high read coverage across the genome to call low frequency variants. Third, read coverage will fluctuate, making reliable estimates of read frequencies difficult. Fourth, variant mis-calls and duplicated regions will complicate the picture.
However, maybe it is possible to pool the genome-wide evidence of many thousand variants to come up with the most likely polyploidy status of your assembly.
if you still have the raw reads (before mapping) or the mapped reads you can try to identify duplicate genomic regions by detecting regions with significantly higher coverage. If your global coverage is of n and some region have a coverage of (theoretically) 2n this might indicate this region is duplicated.
I unfortunately don't have some obvious reference to share (these are just memories from some presentation) but you can check what is done to detect CNVs for example (even though other methods are more widely used).
I hope it has been helpful.
Addition: from the comments below it seems the method is non-trivial and might not be the most suitable one.
Interesting question. In theory, it is not possible to detect a recent, complete auto-polyploid genome from a WGS assembly since the copy number of all chromosomes would scale perfectly with ploidy. That is, if all regions of the genome in a polyploid are the same (ie. no sequence variation among homologous chromosomes), you can't tell if the genome is 1C, 2C, 4C, etc.
However, for an allo-polyploid genome or for partial (auto- or allo-) polyploidy that is not complete across the genome, then it should be possible to detect the polyploidy from assembly of divergent haplotypes or regional differences in read depth as noted by Phillipe.
In plants, there are works suggesting that polyploidization is accompanied by rapid accumulation of mutations (look up Avi Levy's work). So it should be possible to find multiple alleles - heterozygocity for SNPs + indels
Look up Avi Levy's work from the Weizmann Institute of Science (he worked on wheat).
I don't agree with Casey Bergman that "it is not possible to detect a recent, complete auto-polyploid genome from a WGS assembly". Say, your (duplicated) genome is 1Gb in size, you sequence to 100x coverage, so 100Gb. After assembly, for a non-duplicated genome, you would expect the assembly size to be approximately 1 Gb, with an average coverage of the non-repetitive parts approx. 100x. If the genome instead was a complete auto-polyploid (all chromosomes duplicated), duplicate chromosomes collapsed during assembly, so you will see something like a 0.5 Gb total assembly size with an average coverage of the non-repetitive parts of 200x. This is of course an ideal situation, but you get the idea.