1. The thread of the tail you bought is the correct size. The threaded part screws into the radiator after you have removed the existing tail for the union fitting.
2. If you imagine that tail screwed into the radiator, then the plain part sticking out is what the valve fastens on to. The valve fits onto that plain part, and, when you tighten the nut and olive onto it, it seals the valve to the plain part. The nut, shown wrongly screwed onto the part of the the tail which screws into the radiator in your photograph, is used, with another olive, to secure the pipe into the bottom connection of the valve.
3. There are no "union to compression" adaptors. You either have a union tail and a union valve, or a compression tail with a compression valve.
4. You remove the old tail. Then wrap the threaded shank of the new tail with a good number of wraps of PTFE tap and screw it into the radiator. It should screw in until only 1 or 2 threads remain visible, but it must be tight and require reasonable force with a decent spanner. If there are more than 1 or 2 threads showing, remove the tail, remove and discard the PTFE and re-wrap with 20 wraps. There are other ways of fitting tails, using pipe cord (e.g. Loctite 55) or a liquid PTFE like Flomasta Pipe Seal. The liquid PTFE needs the new and old threads to be clean and grease free, not always easy with an old radiator. You can also use gas PTFE tape, which is much thicker than ordinary PTFE and so requires fewer wraps.
5. If you use a compression tail, they can be cut down in length on the plain side to get the valve closer to the radiator. You must leave enough of the plain part to enable the valve to be fully seated on that plain part, plus a couple of mm. The cut must be clean, perpendicular to the axis of the tail and de-burred with a file or similar.
6. If you use a union tail, you will have to change the new valves to union types, which will come with union tails. The valve side of the tail has a convex curve, and the corresponding part of the valve has a matching concave curve.
7. It is always best to use the union valve and tail as supplied. If you use the existing tail with a new union valve, there is a risk that the curves may not be perfectly matched. If you do try this, use a sealant like JetLube V2 to overcome any slight mis-match, and make sure the tail's union surface is perfectly clean.
8. If you do change the tail, you will need a special spanner to remove the old tail. This will have a hexagon inside it, or possible some form of grooving, and the special spanner will fit into this. Screwfix item 133RH is an example of such a spanner, but there other types such as "universal" radiator radiator keys.