Showing £600 for 3 months in your wife's bank a/c would be acceptable (your grandmother 's a/c wouldn't be) but the evidence must be original - see para 87 and onwards in the policy guidance. If it is a printout it would have to be certified by the bank. An alternative would be a letter from your employer - see para 56 in the guidance.
I doubt if the international enquiry service will be of much use (but apart from calling them they also offer a webchat and an email service).
https://ukvi-international.faq-help.com
There would be no harm sending a fax addressed to the Chief Entry Clearance Officer at the British HC outlining your case (bereavement, emergency travel, there was no time to get PBS dependant leave for your UK born son, you are on compassionate leave and have to return to your job asap) - may be they can help in some way
Re. TB testing see:
https://www.gov.uk/tb-test-visa/overview