I work for the NHS and have a contract and job plan. I have 4 young children and use both childcare and parents to enable me to work full time. My job plan states I finish work at 5pm on a Friday afternoon. It takes me 15 minutes to get home. My childcare finishes at 5.15 on Fridays. I have been informed that I must attend a meeting away from work on a Friday afternoon which ends at 5.15pm By the time I travel home it will be 7pm. My childcare cannot stay late on that particular day and every other avenue to help me has been unsuccessful. My employer has made it quite clear that I must attend this meeting. I do not know my rights or what to do as, obviously, my family will always come first. I do need to keep my job though.
Your employer is basically requiring you to work overwork, which you are unable to do on this particular occasion due to childcare reasons. I understand that there is no clause in your contract of employment requiring you to work overtime and therefore your employer is not in a strong position. NB However, your employer could have an argument if you are in a position of such seniority that it is reasonably expected that you should attend the meeting.
In any event, you have made it clear that you have done what you can to seek to make arrangements for childcare, but that unfortunately you cannot work past your contractual finish time of 5pm on this particular day. You have already suggested that you could video conference the meeting and it appears that the medical director has not responded to you about this.
If you were dismissed for not attending the meeting, it is likely that you would be able to succeed with a claim for unfair dismissal and/or sex discrimination.
Given the way in which the medical director has been corresponding with you, one option for you is to lodge a formal grievance. However, if you wish to retain your job and overcome this issue, lodging a formal grievance may not actually assist. It may be better to continue making your points in a professional manner, by email, as you have been doing to date. You could send a further email to the medical director confirming your commitment to your job (you could give examples) and apologising that on this particular occasion you really are unable to attend. You could reiterate your offer to join the meeting by video conference.