I work in the NHS and all of my team have some sort of flexible working agreements in place.
From memory, there is a box on the form our Trust uses to request flexible working that requires agreement that it will be reviewed (usually after 6 months trial and then yearly) to check it still suits the service and can be revoked if not.
There is absolutely no way we would agree a flexible working request that meant not taking at least the legal minimum break during (not at the end of) your shift.
Although informally I know quite a few people including some very senior staff who might occur have children around at home while working, this is something that gets a blind eye turned where people are being productive. Certainly not something we would formally agree to.
Generally, my Trust us very supportive of flexible working but you need to evidence the impact on the service and how it can be mitigated / show any positive impact. We can't demand people give us a reason for their request.
For example - some of my team work compressed hours, we had to show that on their non work days their would still be suitable cover and that there was useful work they could do in the time outside normal hours they were now working. So had to list staff numbers and acknowledge they wouldn't be taking calls but that they could use the time for admin freeing up time in the day for other tasks etc.
I'd really recommend you get a copy of your flexible working policy, speak with your manager and submit a new request. Maybe you could start / finish a little earlier / later Tues-Thurs and then have a shorter day Mon / Fri? Or agree some flexi-time some of which could be worked from home - so maybe core hours of 10-2 or 9-3 in the office and flexi hours between 8-10 2-6 etc. with an agreement these could be worked at home?
Usually your manager would have 28 days to meet with you to discuss your request and you can bring a representative to that meeting.