As I read through the first posts, I was thinking about the FM's stated "take it back!
I thought, WTH, you don't even know if was a dealer buy or private and rules are different
Then the "MOT" bit, too many times people wrongly assume that means its engine is 100 fine, its is NOT.
If the private seller sold after the car was "as seen and or tested/driven" and the seller did not lie - put it down to experience.
A 15 year old car is bound to have problems - op states its a "BMW" could be one on those very little things or it could be one of those flash bigger 7 series and they have eletrics and engines that are as complex as you can imagine.
Older cars and some even newer ones - you sell in good faith but as soon as they new driver, poss their driving style, the cars do come up with a problem to 2
Often a leak looks worrying but at times it may just be a loose bolt,
So if you are buy a car that is over three years old and you see the words "MOTd" - that means the day/time car was tested it was road legal as the law stands and nothing else.
When buying privately, if you know little about cars, take a mechanic, you can buy these services as well from AA/RAC etc
EDIT:
Sorry OP, it does cause worry. But you talk about not having the money to repair it etc. What would you guys done if you had a puncture or two that could not be repaired as decent tyres for cars cost easily upwards of 150 pounds each
Did your OH check the insurance ie costs?
Did your OH check the amount of road tax?
What will happen if the LED headlight fails they cost hundreds
Looks like you OH went out and got a car that looked and ran good but did not think about the, what-ifs
Have you considered selling it?