I use (and have no connection to)
www.transferwise.com very regularly to send money between the UK and US. Routing numbers aren't a problem, the fees are tiny when sending (last time it was <拢6 on an almost $2000 transfer) and there are no fees at all for the receiver. You get the market FX rate too. I think it works by not actuall sending money internationally, but matching up corresponding transfers (or bundles of transfers) both ways. So if I want to transfer to the US, and someone in the US wants to transfer to the UK, my money goes to their recipient, and their money goes to mine, so for the receiving bank it's always a domestic transfer. Other providers that do this are available but like I said I use this a lot. Takes a few days to complete so it's not instant, but it's a lot cheaper and I have no issues with the safety.