Applied late - will my application still be considered?

It's not true.

It might be true that for some courses there will only be a limited number of places left later in the cycle if a university's policy is to make offers throughout, but courses are not kept open once they have been filled. Aside from that, the biggest issue with later applications is that people are likely to have missed funding deadlines.

A quick search showed various people on TSR receiving offers for the courses you've listed despite applying rather late in the cycle. Here, for example.

Thank you for your response.

I did come across that post and messaged the OP to see if he received an offer, but he is yet to respond. As for the person who commented saying he got an offer, I'm not so sure I'm as confident as he was regarding the personal statement

I guess my concern lies with the fact that many people have applied between September-December and therefore the universities have been more generous then due to more spaces available and are only granting offers now to students with an exceptional track record.

Universities have plans in place for how they approach postgraduate recruitment. These plans might look very different for competitive courses at popular universities, and certainly don't involve giving offers to anyone who applies in order to complete the cycle as quickly as possible. You are right that there might be fewer places available now, but I don't think it follows that those places are necessarily harder to secure. Universities also know that they will need to give many more offers than they have places available so someone applying later in the cycle might find they are treated with more leniency because the field of applications has turned out to be weaker than they projected.

This is ultimately all conjecture, and there is no value in me rehashing the various arguments suggesting X and Y because the reality is you have applied now, not in the past. You can be certain that your application will be considered, but all you can do at this stage is wait and see how each university views it. If you are seriously concerned, you can contact the universities and ask when you are likely to hear from them as some have specific dates when applications are assessed. I've made similar enquiries before and found departments quite forthcoming.

Best of luck

That's extremely helpful, thank you. I do hope the bit in bold turns out to be the case

I will definitely consider emailing them and asking, but I'll probably wait a while as it's quite soon.

At this point it is an exercise in patience. Hopefully they process your application(s) quickly!