As above, the MASt is essentially the final year of the 4 year integrated undergraduate masters course. There are in fact thought three Physics-related MASt courses at Cambridge; the Physics one, which has a major project/dissertation which is 1/3 of the total marks for the qualification; the Astrophysics course, which is similar to the Physics one but the emphasis is on Astrophysical topics and they have some overlap with the 4th year (Part III) Maths course for certain relevant options; and finally the Part III Mathematics, which has much less "research" type work, and is mostly exam based, although you can submit an extended essay. For all of them, the exams are noted to be exceptionally challenging.
The Physics course (and department) focuses more on experimental work, although they have some theory in Condensed Matter fields. The astro department is pretty varied, and has both theory stuff in department and via Maths. Most of the theoretical physics work is done in Maths though, which at Cambridge consists of two departments: the Department of Applied Mathematics and Theoretical Physics (DAMTP) and the Department of Pure Mathematics and Mathematical Statistics (DPMMS). They're based on the same site and the division is somewhat more administrative than anything else, but as the title of the former department suggests, it has a major theoretical physics focus (this was the department Stephen Hawking was based in, for example). Part III Maths is the usual prerequisite for a PhD in either of the Maths departments there.
There is then also the MPhil, which will be research based (although you will might well end up going to some of the same Part III lecture courses, just without taking any exams). I would suggest if your aim is a PhD, you should consider what area you would want to do it in - if it's in theoretical physics (except condensed or maybe some astro topics), you should look into the MASt Mathematics course (Part III Maths). If it's in experimental, or maybe condensed matter theory (and not astro-) physics, then the MPhil is probably the best bet. If it's in astrophysics, and theory based, you might want to consider the MASt but otherwise an MPhil is probably more relevant. Do also look at the content of the MASt courses - if you haven't covered material at that level already, you might find it a better fit than the MPhil so you can actually get that background. But, if you are "ready for research" then do so. To my knowledge though either the MPhil or MASt is acceptable to go on to a PhD in either department (and undoubtedly elsewhere). In the UK generally it's fairly typical for students to go frmo a 4 year undergraduate-masters course (e.g. MPhys, MSci, MMath) to a PhD without doing a separate standalone MSc/MPhil (although there are exceptions, like DAMTP at Cambridge, which often expects external students to do the Part III even if they have a masters/undergraduate-masters already it seems).
For Imperial they have a few "flavours" of their Physics MSc, but for all of them you do a several month long research project/thesis over the summer; it is fully a 12 month programme (Oct-Oct, I believe). They also have an extended research version, where you do a 9 month research project in a second year after completing the first year. This latter option is probably most appealing to you. They do however also have their very well regarded Msc in Quantum Fields and Fundamental Forces, which is somewhat similar to the MASt Mathematics at Cambridge in focusing on theoretical physics with some advanced maths that relates to it (e.g. differential geometry, lie groups). The QFFF course however I think assumes a little less exposure to abstract mathematics than Part III at Cambridge. Oxford also have a relatively new MSc in Mathematical & Theoretical Physics, which again is similar to Part III at Cambridge for a theoretical physics masters. Imperial has a dissertation as a requirement, and Oxford has it as an option (being 10% or 20% of your work for Oxford, and I believe more for Imperial). However both, as with Part III at Cambridge, are designed as PhD preparatory programmes.
In terms of UK vs EU masters, in the UK a 1 year MSc is normally equivalent to the two year Bologna format masters in Europe. This is partly due to the fact that, as indicated above, normally such a course is in fact 1 full year - it does not end with exams in May-June and normally you do research in the summer period (although typically you'll begin working on it before then). The MASt programmes at Cambridge are the exception, rather than the rule. Imperial's page for the MSc QFFF for example specifically notes it is Bologna compliant and acceptable to continue to a Bologna format PhD in the EU.