Employer not paying TAX or NI

Did the employer makes those threats verbally, or via text or other written medium?

The employer is required by law to provide a P45. Your partner can report to HMRC that he has not had one and they can take action, and when he get s a new job he will need to ask his new employer for a 'starter checklist' which allows them to get his tax code right.

He could issue a claim for the unpaid wages. I think that reclaiming the petrol costs might be harder depending on whether he ever got a written contract and what it said about reclaiming expenses (and whether he could prove the petrol was for a work vehicle, not his own)

In terms of the employer's threats:
- Just becuase he threatens to send an invoice, doesn't mean he will do so.
- just because he sends an invice , doesn't mean that your partner has to pay it
- if he sought to claim the money he could make a claim via the small claims court, and your partner could defend it. The employer would have to be able to prove that your partner damaged the vehicles.
- If he sought to claim the cost of an agency worker he would, at most, be able to claim actual costs as a result of your OH leaving without notice. If there was no written contract of employment, then your OH would only have had to give one week's notice. The employer would have to prove that they had in fact incurred the expense which, if they did not, in fact, take on another worker, they won't be able to do.

So, in your partner's place I would:
1. Write to the employer (keeping a copy and getting proof of postage) requesting that they provide the P45, outstanding wages (+ Pay for any holidays accrued but not taken) and repayment of expenses.
2. Contact HMRC, notify them of the date he finished work and that the employer is refusing to provide a P45. (also, if necessary, supply them with the wageslips as proof of income and tax)
3. If no payment is forthcoming, go to money claim online and follow the instructions to make a claim. Make sure that you check first whether the employer is a limited company or a sole trader / partnership so you sue the correct legal entity.

Do 2 even if you decide not to do 1 or 3.

There is also nothing to stop you contacting the police and/or DVLA about the concerns relating to untaxed vehicles.