Hi Jude
Most means-tested benefits are only paid to people with less than £16,000 in savings (tax credits and the guarantee element of Pension Credit are not affected by this rule).
If you (and your partner) are under Pension Credit age and have total capital over £6,000 then the government assumes you receive £1 per week for every £250 of capital you have above £6,000.
If you are over Pension Credit age the government assumes you receive £1 per week for every £500 of capital or part over the £10,000. There is currently no upper limit, you may have deemed income from capital. What this means is if you have £12,005 capital you will be treated as having £5 a week income.
In answer to your question, If you are in receipt of CB ESA then you will not be affected by any savings, if you transfer to UC, this is means tested and you will be affected by any savings over £6k.
Gary
Nothing on this board constitutes legal advice - always consult a professional about specific problems 
