Getting into REPE UK - Non-traditional, non-target

I come from a totally non-traditional background, I study Chemical Engineering and I managed to get onto a sponsored degree course (i.e. employer pays Uni fees and I get paid a salary to attend University part-time as well as working) with a large corporation. University is non-target though I had the grades and I was meant to go to a target school before accepting sponsored offer. 

I spent some time working in land acquisitions for a housebuilder as work experience when I was a teenager and absolutely loved it, which has led me to consider working in RE since I can't see myself doing my current role for my entire career. I would love to get into REPE, but given my background I know I'll struggle as I a) have no relevant experience b) did not go to a target school. In terms of getting a foot in the door is there anything I can do to leverage my alternative experience or to get into more traditional routes such as internships once I have completed my degree?

I'll probably need to complete a masters, are there any recommended courses I should consider other than Cambridge real estate finance, LSE finance etc? Maybe Henley, I know Uni of Manchester also offer a Msc Real Estate course but not sure how this is viewed in real estate finance circles?