What is the foundation of the claim that the British robbed India of $45 trillion and caused the deaths of 1.8 billion Indians during their rule? Are these figures accepted by modern scholars?

I'm sure there are several appropriate threads on the subreddit which have addressed this question, however, I would like to offer my two cents, for what that may be worth. We begin first by looking at how Prof Usha arrived at the numbers that she projects. According to the article that you yourself have quoted :

"Professor Utsa Patnaik has estimated the magnitude of the British robbing of India thus:

Between 1765 and 1938, the drain amounted to 9.2 trillion pounds ($45 trillion), taking India’s export surplus earnings as the measure, and compounding it at a 5 per cent rate of interest."

This methodology itself is highly questionable, however, the conclusions that it seeks to arrive at, show that, it follows in the footsteps of a tradition of nationalist-Marxist historiography offset by such historians as Dadabhai Naoroji and Romesh Chandra Dutt, which has attempted to quantify the economic damage suffered by the South Asia (for convenience referred to as India from here on out), if indeed such a thing could be quantified.

The argument discernible from the methodology quoted above is certain. Britian siphoned off the profits accrued owing to export surpluses generated by it's Indian colony, in essence, "robbing the colony". Though the numbers may seem fantastic, Irfan Habib in his, 'Studying a Colonial Economy - Without Perceiving Colonialism', Modern Asian Studies,1985, pp. 375-6, quotes a figure to the tune of Rs. 1,355 million for the drain in 1881, in 1946-47 prices.

The numbers have varied and so have the methodologies. Usually the attempt has been to portray the expenditure of the Indian government for financing public debt, military forces, infrastructure, payment of remittances etc. were unjust "robbing" of the colony's resources and revenues, which rightfully belonged to locals, or Indians.

We can first look at how the number has been arrived at, namely the figure of $45 trillion. Here I'll be using quotes by Prof. Patnaik, in her own article, in the "Monthly E-Newsletter – Amity Business School, Volume V, Issue I, October 18", titled "How the British Impoverished India"

After decades of research I find that using India’s commodity export surplus as the measure and applying an interest rate of 5%, the total drain from 1765 to 1938, compounded up to 2016, comes to £9.2 trillion; since $4.86 exchanged for £1 those days, this sum equals about $45 trillion

There is first the issue of using export surplus as the measure for appropriation. Before approaching which, I must draw attention to the seemingly arbitrary 5% interest rate, for which we are given no explanation. There is then the use of exchange rates from pre-1938, passed off as $USD in 2016, expressed in terms of it's value in comaprison to the £ in the same era, while the exchange rates for the two currencies were drastically different in 2016. This is, not reflective of serious thought being applied to this evaluation. To add to that, according to Tomlison, B.R.The New Cambridge History Of India III, The Economy of Modern India, 1860-1970, 2003, p. 13-14

The size of the unrequited transfers, those needed to meet the 'Home Charges' (the administrative and military expenses of the Indian Government in Britain), was small, running at around Rs 20 million a year, less than 2 per cent of total export values at the end of 1913.

[ Source : India's International Economy in the Nineteenth Century: An Historical Survey Author(s): K. N. Chaudhuri, Modern Asian Studies, Vol. 2, No. 1 (1968) ]

Prof Patnaik sheds light as to how she arrives at this conclusion in the article :

The Secretary of State for India in Council, based in London, invited foreign importers to deposit with him the payment (in gold, sterling and their own currencies) for their net imports from India, and these gold and forex payments disappeared into the yawning maw of the SoS’s account in the Bank of England. Against India’s net foreign earnings he issued bills, termed Council bills (CBs), to an equivalent rupee value.

Description of such bills of exchange as "appropriation" by the British government, is highly contentious and doesn't seem to be based in any evidence. The sale of Council Drafts was a convenient tool for the management of Indian currency, exchange, and finance. Usually, India ran a surplus on trade account, yet, payment had to be made in sterling to meet the ‘Home Charges’. The Secretary of State in Council invited tenders for delivery of sterling in London against payment in rupees from Government funds in India. The Act XVII of the year 1835, declared the silver rupee of 180 grains troy, 11/12ths fine, as the sole legal tender currency of British India. Furthermore, average production of silver rose from 1.34 million kgs. during 1866-70 to 1.97 million kgs. during 1871-75 and further to 2.86 million kgs. during 1881-85, gold output declined from an average of 0.20 million kgs. during 1866-70 to 0.17 million kgs. during 1871-75 and further to 0.15 million kgs. during 1881-85. This severely depreciated the value of the Indian rupee. To add on to the problems of the Government of India, the payments for the "Home Charges" of the Government, being interest on debt, pensions, payments to the War Office, cost of Government stores, etc. increased, and these had to be made in sterling. The government flirted with a gold standard, and a committee was setup for the same purpose under Lord Herschel, the Lord Chancellor, in October 1892 by the Secretary of State, yet the growing silver depreciation, explain the necessity for the Council Drafts as a means of exchange.

[ Sources :

Transition from Indian to British Indian Systems of Money and Banking 1800-1850 by Amiya Kumar Bagchi, Modern Asian Studies, Vol. 19, No. 3
Money, Prices, and Economic Development in India, 1861-1895. John Adams and Robert Craig West, The Journal of Economic History, Vol. 39, No. 1]