Plumbing – Cap galvanized water pipe

It's very difficult to tell what your existing pipe is like, as there's a shower attached to it. However, I think your problem is that the existing pipe diameter is larger than the UK shower connection? In which case you need a reducing coupler.

Here's an example. But a) your existing pipe may be 1" rather than 22mm; and b) the fitting at the larger diameter end may not be suitable. However, if it were that simple, then your plumber should have been able to do it. So I suspect problem a). I'm sure somewhere must sell imperial to metric reducing couplers, but I haven't looked.

With regards to having an ugly converter sticking out the wall - your other option is to have the converter behind the tiles. This requires either 1) breaking out the tiles, then retiling a small area afterwards; or 2) coming at the wall from the other side (in which case you'd then need to replaster the other side of that wall). I personally did 2 when installing a new shower, as the tiles were installed by the previous owner, so I didn't have any spares, but it had been done recently so I didn't want to retile the whole bathroom. Option 1 shouldn't be too difficult, as you say you have retiled, so you should have access to identical replacement tiles.

Comparing the pressure drop between 50' of 3/4" and 1" PEX...going up to 1" saves 1 psi at 5 GPM flow rate, and 5 psi at 9 GPM and 7 psi at 11 GPM (when you hit 10 feet per second in 3/4" PEX, a velocity which shouldn't be exceeded). In other words, you need to have pretty significant concurrent demands before the pressure drop in that pipe run becomes noticeable (or where pipe erosion from high velocity becomes a concern) - like filling two tubs or taking four showers at the exact same time, plus running a dishwasher or washing machine. It's possible that with so many kids that's actually a reasonable concern for you, but for the typical household there would be little benefit.