There is a large weld on the exhaust where the DPF is/should be. I had a mate put the car through an MOT emissions test. It passed easily.
There is no MoT emissions test for diesels. If there was one, that would force the emissions equipment (and that includes egr, dpf, etc) to be the same as the original model acceptance tests. Well, to the extent that the changes are detectable, of course.
There is a smoke test and that just tests smoke, not oxides of nitrogen or sulphur or carbon. So, if the smoke levels are low, that's a pass (on that part of the test...you could fail on other parts, of course).
Normally, you would expect to see some kind of box in the exhaust that contains the DPF. I would guess that you mean a piece of pipe has been welded in where this may once have been. As far as I remember, this is no longer legitimate, and you could be forced to put a box back in (there isn't really a test to check how effective this box is, so it could, in theory, be an empty box, provided that the vehicle meets the smoke test, which they will often do).
As a new reg and one that is difficult, in extremis, to test against, some MoT examiners may be more liberal than others, but there is clearly a danger of failing this part of the test. OTOH, depending on what exactly has been done and whether the engine has had some change which would need to be undone, it might be easy enough to revert, provided that you have a DPF and welding equipment.
After noticing a bit of smoke from the exhaust a garage said this was normal and I should burn it off by travelling at high revs for a short while.
Another thing is I have never seen my heating coil light on the dash. Thought I always should.
Am I worried about nothing and I just have a light out? Has my DPF been removed and this is allowing soot to build and blow out of the exhaust?
It seems that is bad advice from the garage. If the DPF is present, soot can build up in the DPF that is then burnt off during a period of sustained high temperatures. However, if, as you seem to be saying, the DPF box has been removed, you can't burn the soot from the DPF box with an extended run, because you don't have the DPF box.
If the DPF has been removed, soot will not be building up because it would primarily be building up in the DPF box. It will be blowing out of the exhaust because there is no DPF box. It won't be being rendered harmless (well, less harmful) by being burnt, because that would happen in the DPF box, so you will still be murdering people (but only a little bit each, if that helps).
When you initially 'key on' there should be a brief lights tests. Any dash lights that don't respond have been sabotaged in some way (bulb removed, disconnected, whatever). Beyond that, you should only get the glow plug lights on if relevant; ie, it is cold. When the car has been recently run, or if the ambient conditions are not cold, this will not be necessary as the car will start without them.