i suspect the podiatry department of hospitals may be much busier than the ones at polyclinics. if you can wait... go to your company's GP, ask for a referral letter to a polyclinic podiatrist.
i've been to the one at bukit batok polyclinic. call the polyclinic up directly (URL below) to make an appointment. be sure to let them know you have a referral letter. call around to find one that has the shortest wait before committing to an appointment.
as i wasn't in singapore, the soonest from the time i called until i saw the UK-trained expat podiatrist at the polyclinic was about 4 weeks. the orthotics took another 3 weeks or so (i can't remember) to build.
be sure to ask for a sweat proof cover/material for the top of your orthotics so they last longer. tell the podiatrist you have sweaty feet. bring along whatever shoes you want to wear them with as well - it's useful for him/her when making your prescription.
before this visit, i used to go to the foot clinic (used to be operated by TTSH but has since closed) and 2 private podiatrists... a polyclinic's cheaper than going private, and you get to keep the plaster cast of your feet for future use (some podiatrists in private practice keep the cast to ensure you go back to them).
if you're like me, you probably want to make a few pairs at a go... i can't remember the cost breakdown for the cast and each pair, so call and ask.
singhealth: Geylang & Tampines
nhg: Bukit Batok, Hougang, Jurong, Toa Payoh, Woodlands & Yishun
http://www.podiatry.org.sg/members/