There shouldn't be problem but you'll be looking at 2 sets of legal fees which may eat into any savings you may make. Check with the lenders first, there may be a condition that you cannot switch for 6 months, of course if you fix and break there will be a penalty.
Always be wary of the lenders who offer different rates for new and existing business. The banks never disclose this at application stage.
But there would be no guarantee that the original lender would offer you a new mortgage?
I can understand why in the days of mad competition, lenders offered new customers better rates. I don't understand why they would do this these days?
In fact, they probably should be doing the very opposite. If there is discretion regarding rates, keeping them lower for existing customers would be good business.