VAT
To clarify re: VAT...it's your taxable supplies that matter. Exempt supplies
are not taxable supplies, zero rated supplies
are taxable supplies, they just so happen to be taxed at zero percent.
So you need to clarify whether your sales are zero rated or exempt. I'm not an expert on the tax status of nappies (!) but I know children's clothing is zero rated, so quite possibly nappies are also under the same bracket.
You can have exempt sales of millions of pounds and not register for VAT, it's only your taxable supplies that count (including standard rate, reduced rate, and zero rate).
Typically if your supplies are zero rated, then it's normally best to register for VAT voluntarily, cos you can still reclaim all your input VAT. In your case, assuming your simply reselling the nappies, most of your purchases are probably zero rated too, so it won't have that big an impact, but you will be able to reclaim VAT on standard rated overheads
Overtrading
I wouldn't really have thought this is what your accountant was referring to when he said you run the risk of overtrading though.
An extreme example of overtrading is a small start up business which does really well at selling. They sell loads, and have a full order book. Their customers (say) are invoiced on delivery and expected to pay within 30 days. However, as the business has minimal track record, the supplier demands payment up front. The business then has to borrow money to pay the supplier to fill the orders, then if customers don't pay/are slow to pay the business can't repay their borrowings and go bust despite having huge sales and huge (paper) profits.
In short, it comes back to the old mantra "turnover is vanity, profit is sanity, cash is reality". Ie sales don't mean nothing if you ain't making a profit, and a paper profit don't mean nothing if it doesn't turn into cash.
If (as has been suggested above) your customers typically pay cash up front, I don't really see what the problem is, might be worth getting your accountant to clarify what he thinks may be risky.