I would advise exercising caution here as this seems to be one of those regrettable situations where the person causing the trouble has more rights than his victim.
Our questioner has quite rightly identified two potential problems with a DIY removal - hence the suggestion of using a private company to remove it.
1. How is access to be effected? If he/she breaks into the car then a crime will have been comitted however much justification there appears to be - probably vandalism.
2. If the questioner steers or even simply pushes the car onto the public highway in order to then inform DVLA, then there is a possibility that our questioner will be charged with theft, and then driving/being in control of a vehicle on a public highway without insurance, road tax, or MOT.
I think that this issue needs to be solved using the Council in the first instance. If they won't remove it for whatever reason, then I think this would need to be removed after getting a Court order - so you'll probably need to employ the services of a solicitor for that.