If they're paying you a mileage (or is it kilometreage in the UK?) reimbursement, then no, you shouldn't try to get reimbursed for the tire. The mileage rate is supposed to cover all expenses associated with operating the car, including the portion of maintenance and repairs that the work trip contributes to. You just got unlucky that the tire blew when you were traveling for work instead of personal reasons.
If, however, they are only reimbursing for actual fuel expenses, then you might try to get reimbursed for the tire. But I suspect they are under no obligation to reimburse for it. Does your company have a policy on use of personal vehicles for work? Read it and see what they say about such expenses.