Diners is by far the better conversion tool, only burning 50% of your points vs. 75% of your points with Hilton.
You can convert AA & UA points through Diners and presuming that you inded to do it now you would burn half those miles in the conversion. As well, you can only deposit UA or AA miles in 10000 increments to Diners, so you could only put in 20K AA points and earn 10K BA miles. If you can wait till the summer there is NO burn from this transfer under the Diners BA double miles promotion.
Good luck