Calls from abroad, using mobile phones, generally don't cost much these days. For example Vodafone only charges me 4p per minute when calling from Europe (despite being on a PAYG tariff, using my phone for emergencies only, which charges me 30p per minute in the UK).
I suspect that the doctor's appointments system is configured to only accept internet traffic with an IP address based in the UK. To get into it from abroad you'd need to use a VPN which routes your web traffic via a UK proxy server. You've not said which device you're using (Windows laptop? Android phone? etc) but this freebie is popular with users of Windows computers:
http://expat-shield.en.softonic.com/