In order to put the mortgage on the first house into joint names you will probably need to transfer the property into joint names, which isn't a bad idea in general, no tax implications there, income from the property will be assesed in the same way as your other income. If the mortgage is over 254K you will have to pay stamp duty at .1% on mortgage.
Either option will work but doing a top-up may work out cheaper, but you will have to get independant legal advice and sign a lot of consents and depending on the bank possibly guarantee the mortgage anyway.
You should talk to your solicitor before deciding but tax isn't really an issue. You won't qualify for first time buyer rate on the new house even if you buy in your own name for various reasons that I wont bore you with, I suspect in any event that the new house is over the limit for a stamp duty concession.