I think talking about deposits is confusing when you already own a property, you actually need to think about equity. Your equity is the value of the house minus the remaining mortgage, so to give an example your current property is worth £180k and the outstanding mortgage amount is £130k say, so you have £50k equity.
For the new purchase, you then need a mortgage for the purchase price less your equity (unless you want to put in additional savings to reduce the mortgage amount or keep some of the equity as cash). But assuming the new house costs £280k, and you want to put it the full equity from your sale, you need a mortgage for £230k. You don't need to put up a 'deposit' as such because the equity is coming from the sale, assuming you are able to get a mortgage for £230k of course. If you could only borrow £225k then you'd need to pay the additional £5k from savings.
You do also need to make sure you have cash to cover the sale and purchase costs (solicitors fees, survey cost, estate agents commission, removal costs, any building work you want done immediately on the new house) and also the stamp duty, unless you want to take these costs out of your equity, in which case you need a bigger mortgage - say you were allowing £10k to cover these costs, you'd need a mortgage for £240k - but if you have the savings definitely better to pay the costs upfront as you'll pay more interest on a bigger mortgage.
Hope this makes sense!
Do be aware also of whether your current mortgage has any early redemption fees in which case as PP says you will need to stay with the same provider...