It would most likely fail an MOT without a DPF, now it is not illegal to remove them but it is illegal to drive around without one fitted if it was part of your cars manufactured specification, and it's also a modification that you could not get insured on, so technicality your insurance is null and void! Also if you were the car behind would you want to breath in all of the particulates coming out of your engine.
Personally I would ditch the diesel and look around for a petrol, if you only doing 7 mile runs the engine is not getting up to temperature, and these short trips don't do an favors for TDI engines as they never get up to the optimal running temperature.