What is the is the difference between mechanical insurance policy underwritten and administered by an insurance company

The best definition is from the Insurance (Prudntial Supervision) Act which says

Section 7 Meaning of contract of insurance

(1) For the purposes of this Act, unless the context otherwise requires, contract of insurance—

(a) means a contract involving the transference of risk and under which a person (the insurer) agrees, in return for a premium, to pay to or for the account of another person (the policyholder) a sum of money or its equivalent, whether by way of indemnity or otherwise, on the happening of 1 or more uncertain events; and

(b) includes a contract of reinsurance.

(2) In this section, uncertain event means an event—

(a) with respect to which there is (from the perspective of the policyholder) an element of uncertainty as to when or whether it will take place; and

(b) that is beyond the insurer’s control.

(3) However, a contract, to the extent that it provides for, or relates to, any of the following is not a contract of insurance for the purposes of this Act:

(a) a derivative transaction:

(b) a guarantee under which a person agrees to answer to another person for the debt, default, or liability of a third person:

(c) a repayment waiver (within the meaning of section 5 of the Credit Contracts and Consumer Finance Act 2003):

(d) a product or service guarantee or warranty in relation to any goods or services that is given or made by the manufacturer or supplier:

(e) any lump sum, annuity, pension, allowance, refund, or other payment arising from membership of a retirement scheme (within the meaning of section 6(1) of the Financial Markets Conduct Act 2013):

(f) gambling (within the meaning of section 4(1) of the Gambling Act 2003):

(g) a call-out service that involves providing assistance in respect of a motor vehicle (within the meaning of section 2(1) of the Land Transport Act 1998) that cannot be operated (for example, because it has broken down, has run out of fuel, or has a flat tyre, or because the owner or operator is locked out), being a service that is not provided when the motor vehicle is involved in a motor vehicle accident, is stolen, or is damaged owing to theft or vandalism:

(h) any other transaction or matter of a class declared by regulations to be transactions or matters that are not by way of insurance.

If your scheme falls into Section 7(3)(d) it may not be insurance

Essentially you are trying to create a contract which says to the car buyer that they will be covered for WOF, service costs, breakdown and repairs. That is a situation where the buyer is entering a contract to transfer some risk, which then does make it insurance. It is uncertain what will be needed for the WOF each time it is due, and similarly for the service. This means there must be a transfer of risk to the dealer.