I had the same thing happen with a side-shift trans; internally, the two (2) gearboxes are nearly identical, only differing in detail and ratios.
In my case, the third-gear synchronizer was too thin, allowing the center of the synchromesh hub to move too far rearward while its collar moved too far forward to engage fourth gear. As Steve says, the three (3) balls in the hub escaped, and the three (3) springs that project out of their recesses are what's blocking the collar to move, holding it in fourth gear. IF that's the case with your trans, you will never get all three (3 each) springs and balls in their recessed bores until the trans is disassembled.
It could be worse; it happened to me while driving from Atlanta, Georgia to Spring Hill, Florida__over four hundred (>400) miles using 4th & 4th/ovd! Fortunately, the Healey has a lot of torque, and by retarding the timing until it didn't ping, I was still one of the quickest cars leaving the stoplights! Trick was to not slip the clutch any more than to get the car rolling, and let it lock up as quickly as possible, letting the torque do all the work.
Upon arriving at my bother's place, and using him to help me get the trans out, a trip to the local ACE Hdwr provided suitable replacement springs, and swapping the 3rd gear synchro for the 4th gear one on reassembly. I only lost a full day of my trip, and only because I had to wait for my brother to get home from work both days, to help get the trans out and back in (this taking place in the mid-80s, less than a year after breaking my back).
The 3rd/4th gear synchro hub being assembled:
You can follow through the sequence of pictures, beginning on the page linked below, to see how it's assembled and mates with the other components:
(the hub thumbnail-pictures are the 3 on the bottom row, and continued on the following page)
https://spcarsplus.com/gallery3/index.php/Healey-at-home/gearbox_work/sideshift_xmsn_assembly?page=8