I would note that the OU grading doesn't correspond to "brick and mortar" uni classifications - the equivalent to a 1st class result in an OU module would be a pass grade 1/distinction (usually 85% or more). I imagine pass grade 2 would be the minimum expectation for Oxford, if not distinctions. A potentially more significant issue is that the OU may not provide the necessary depth in certain areas of pure maths compared to those who have undertaken a maths degree elsewhere, which may be an issue for either programme ("even" if you want to specialise in applied or applicable maths, because core areas of pure maths necessarily underpin these, mostly analysis).
While between the two pure maths modules you cover the essentials of linear algebra, basic analysis and some metric spaces (among other material less relevant to applicable and applied maths), the structure of the modules and assessment as I understand is more in common with the earlier modules offered e.g. multiple choice exam and short problem sheets with a lot of computational elements. There seems to be less formal proof writing in either the TMAs or exam for those modules, compared to brick and mortar uni students who would be taking modules covering similar material and having them be essentially all proof based, to some extent, which may not really provide you with the necessary background to undertake the lecture courses offered on the respective masters programmes at Oxford and Cambridge.
In theory it would be possible but I don't think you will probably be able to directly apply to a masters course at either, realistically, and I think you would probably need to do some further study beforehand. KCL used to offer a standalone postgraduate diploma programme that let you take modules from years 2/3 of their undergraduate programme, which probably would be an ideal way to bridge any potential gap in background. However I can't find info about it on the "new" KCL site, so it's unclear whether the programme has been retired as a standalone qualification and the PGDip is now only offered as an interim award if you fail to complete the masters there...It's worth noting that Part III at Cambridge specifically indicates (or at least previously did) that it is not uncommon for students (particularly from education systems dissimilar to the UK brick and mortar uni format for maths degrees) to apply to it already holding a masters degree in maths.